A Study of ASP1012 in Adults With Solid Tumors

Inclusion Criteria: * Participants in Parts 1 and 2 must have histologically, or cytologically, confirmed diagnosis of locally advanced or metastatic solid tumor(s). * Dose Escalation (Part 1) - all previously treated participants with solid tumor types * Dose Expansion (Part 2) * Participants with previously treated cutaneous melanoma, that is, anti-programmed cell death protein 1 (PD-1)/programmed death ligand 1 (PD-L1) alone or in combination with anti-cytotoxic T-lymphocyte-associated protein 4 (CTLA-4) inhibitor. Participants with BRAF-mutant melanoma must have received a v-raf murine sarcoma viral oncogene homolog B (BRAF) inhibitor as monotherapy or in combination with other targeted agents (for example, murine embryonic fibroblasts (MEK) inhibitors). * Participants with previously treated solid tumors, * Participants with stage IIIB to IIID or oligometastatic resectable stage IV, treatment-naïve melanoma that are amenable to resection. Note: Participants with acral lentiginous melanoma will be excluded. * Participants in Part 3 (Dose Expansion) must have histologically, or cytologically confirmed diagnosis of either: * Previously treated (at least 1 line of prior therapy) gastric cancer (including gastroesophageal junction cancer \[type 2 and 3 only\]). Prior lines of therapy may include combination chemotherapy such as FOLFOX based regimen containing 5-flurouracil, leucovorin and oxaliplatin, targeted therapies against human epidermal growth factor receptor 2 (HER2+) tumors and checkpoint inhibitors. * Stage II to IV CRC in complete remission with no measurable disease as defined by RECIST v1.1 following surgical resection and adjuvant therapy with circulating tumor deoxyribonucleic acid (ctDNA) detectable by local testing; the ctDNA positivity will be confirmed during the trial by centralized testing Note: Participants with Stage IV CRC are limited to those with oligometastatic disease in liver. Participants with CRC must have received and completed standard of care and adjuvant therapies which may include fluoropyrimidine, oxaliplatin, bevacizumab, and irinotecan-based chemotherapy and surgery. * Stage II to IV ovarian cancer including breast cancer gene mutations in complete remission with no measurable disease as defined by RECIST v1.1 following surgical resection and adjuvant therapy with CA-125 concentration exceeding 2 times of normal level (\> 70 units per milliliter \[U/mL\]) as measured by local testing. Participants with ovarian cancer must have received standard of care and adjuvant therapies which may include platinum-based chemotherapy and/or poly-ADP ribose polymerase (PARP) inhibitors. * Other solid tumor type (when identified), for example, a tumor type in which antitumor activity or biomarker response is observed in Parts 1 or 2 or additional tumor types of interest. * Participant has measurable disease as determined by RECIST v1.1, except for participants with CRC and ovarian cancer enrolled in Part 3. Lesions situated in a previously irradiated area are considered measurable if progression has been demonstrated in such lesions. * Participant has progressed, relapsed or discontinued for toxicity during or after the last systemic anti-neoplastic therapy and is unlikely to achieve clinical benefit from standard of care therapies per investigator, except for Part 2 participants with treatment-naïve melanoma and Part 3 (participants with CRC and ovarian cancer). There is no limit to the number of prior anti-neoplastic therapies received. * Participant has a predicted life expectancy ≥ 12 weeks. * Participant has at least 1 site of disease suitable for biopsy (except for Part 3 \[participants with CRC and ovarian cancer\]) and is willing and able to undergo required tumor biopsies according to the treating institution's guidelines at screening and during study treatment. Bone biopsies are not acceptable. * Participant has an ECOG performance status of 0 or 1. * Female participant: * Is not pregnant and at least 1 of the following conditions apply: * Not a woman of child bearing potential (WOCBP) * WOCBP who has a negative urine or serum pregnancy test at screening and agrees to follow the contraceptive guidance from the time of informed consent throughout the treatment period and for at least 180 days after the final ASP1012 administration. * Must not be breastfeeding or lactating starting at screening and throughout the treatment period and 180 days after the final ASP1012 administration. * Must not donate ova starting at first administration of study intervention and throughout the treatment period and for 180 days after the final ASP1012 administration. * Male participant: * Must agree to use contraception with female partner(s) of childbearing potential (including breastfeeding partner) throughout the treatment period and for 180 days after the final ASP1012 administration. * Must agree to remain abstinent or use a condom with pregnant partner(s) for the duration of the pregnancy throughout the investigational period and for 180 days after the final ASP1012 administration. * Must not donate sperm during the treatment period and for 180 days after the final ASP1012 administration. * Participant must be willing and able to comply with the study requirements including prohibited concomitant medication restrictions. * Participant agrees not to participate in another interventional study while receiving ASP1012 in the present study/participating in the present study. Exclusion Criteria: * Participant has ongoing toxicity ≥ Grade 2 per the Common Terminology Criteria for Adverse Events (CTCAE) version 5.0 considered clinically significant and attributable to prior anti-neoplastic therapies. Note: Participants with peripheral neuropathy, type 1 diabetes mellitus, endocrinopathies stably maintained on appropriate replacement therapy or skin disorders (for example, vitiligo, psoriasis, or alopecia) not requiring systemic treatment are allowed. * Participant has had major surgery ≤ 4 weeks of signing the informed consent form (ICF). * Participant has symptomatic or untreated central nervous system metastases or leptomeningeal disease. Participants with treated symptomatic brain metastases should be neurologically stable (for 4 weeks post-treatment and prior to screening) and off steroids for at least 2 weeks prior to first administration of ASP1012. * Participant has active or prior autoimmune or inflammatory disorders requiring systemic therapy within the past 2 years (including inflammatory skin conditions, or inflammatory bowel disease \[for example, colitis or Crohn's disease\], diverticulitis \[with the exception of diverticulosis\], celiac disease, systemic lupus erythematosus, non-infectious pneumonitis, Sarcoidosis syndrome, Wegener syndrome \[granulomatosis with polyangiitis\], Graves' disease, rheumatoid arthritis, hypophysitis, uveitis). The following are exceptions to this criterion: * Participant with type 1 diabetes mellitus * Participant with vitiligo or alopecia * Participant with endocrinopathies stably maintained on appropriate replacement therapy. * Participant with any chronic skin condition that does not require systemic therapy * Participant with eczema or history of eczema requiring systemic treatment. * Participant with another malignancy that currently requires treatment * Participant with a condition requiring systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of first administration of ASP1012. Inhaled or topical steroids and adrenal replacement doses ≤ 10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease. * Participant has received a prior allogeneic bone marrow or solid organ transplant. * Participant has a condition requiring use of anti-viral agents with a potential to inhibit vaccinia replication. Prophylactic use of anti-viral agents (e.g., acyclovir) is permitted. * Participant has an uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, any form of substance abuse or psychiatric illness/social situations that would limit compliance with study visits or requirements or a condition that could invalidate communication. * Participant has a known history of human immunodeficiency virus (HIV) infection or suffers from other acquired and congenital immunodeficiency diseases. HIV testing is not required for the purposes of this study unless mandated by local health authority. * Participant has a history of moderate to severe ascites, clinically significant and/or rapidly accumulating ascites, bleeding esophageal varices, hepatic encephalopathy or pericardial and/or pleural effusions related to liver insufficiency. * Participant has a clinically significant abnormal ECG at screening that imposes a safety risk for participation in the study. * Participant has symptomatic cardiovascular disease within the preceding 12 months including but not limited to the following: significant coronary artery disease (for example, requiring angioplasty or stenting), acute myocardial infarction or unstable angina pectoris \< 3 months prior to screening, uncontrolled hypertension, clinically significant arrhythmia or congestive heart failure (New York Heart Association grade ≥ 2), pericarditis or myocarditis \< 3 months prior to screening. Prior/Concomitant Therapy * Participant has received a live vaccine against infectious diseases or COVID-19 vaccine within 28 days prior to the first dose of ASP1012. If a participant has received Paxlovid for COVID-19, a 1-week washout period is required prior to the first dose of ASP1012. * Participant has received prior radiation therapy within 2 weeks of the first dose of ASP1012. Participant must have recovered from all radiation-related toxicities, not require corticosteroids and not have had radiation pneumonitis. A 1-week washout is permitted for palliative radiation (≤ 2 weeks of radiotherapy) to non-central nervous system disease. Prior/Concurrent Clinical Study Experience * Participant is concurrently participating in another interventional study or has received an investigational product or other immunotherapy anti-cancer therapy or anti-viral therapy ≤ 30 days or 5 half-lives, whichever is shorter, prior to first administration of ASP1012. Diagnostic Assessments * Participant with inadequate organ and marrow functions meeting any of the below criteria: * Leukocytes \< 3000/microliter (µL) * Absolute neutrophil count \< 1500/µL (for participants with known Duffy null or benign ethnic neutropenia, absolute neutrophil count \< 1000/μL) * Platelets \< 100 000/µL * Hemoglobin \< 9 grams per deciliter (g/dL) * International normalized ratio (INR) \> 1.5 × upper limit of normal (ULN) and/or activated partial thromboplastin time \> 1.5 × institutional normal limits total bilirubin (TBL) \> 1.5 × institutional normal limits (participants with known Gilbert syndrome who are excluded if TBL \> 3.0 × institutional normal limits or direct bilirubin \> 1.5 × institutional normal limits) aspartate aminotransferase (AST) and/or alanine aminotransferase (ALT) \> 3.0 × institutional normal limits; Participants with tumors in the liver, AST and/or ALT \> 5 × institutional normal limits * Albumin \< 2.5 g/dL * Creatinine clearance \< 45 milliliter per minute (mL/min) (calculated by Cockcroft-Gault equation). Alternatively, GFR \< 45 mL/min/1.73 m\^2 (calculated by Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) equation) can be used. * Participant has any of the following per screening serology test: * Hepatitis A virus antibodies (immunoglobulin M) * Positive hepatitis B surface antigen (HBsAg) or detectable hepatitis B virus (HBV) DNA. Participants with negative HBsAg, positive hepatitis B core antigen and negative hepatitis B surface antibodies are eligible if HBV DNA is undetectable. * HCV antibodies unless hepatitis C virus (HCV) ribonucleic acid (RNA) is undetectable * Positive coronavirus 2019 antigen test 10 days prior to cycle 1 day 1 (C1D1) with ASP1012 Other Exclusion Criteria * Participant has any condition, which makes the participant unsuitable for study participation. * Participant has a known or suspected hypersensitivity to ASP1012 or any components of the formulation used, including prior adverse reaction to vaccinia (for example, as smallpox vaccine). * Participant has had previous exposure with ASP1012 or known exposure (\< 5 years) to pox virus.