Avelumab and Trabectedin in Treating Patients With Liposarcoma or Leiomyosarcoma That is Metastatic or Cannot Be Removed by Surgery

Inclusion Criteria: * Must have a histologically confirmed diagnosis of advanced (metastatic or unresectable) soft tissue sarcoma with one of the following subtypes: * Leiomyosarcoma * Liposarcoma * Subject must be clinically indicated to receive trabectedin therapy as part of routine care. Subjects may be first line, or have received any number of prior systemic therapies * Total bilirubin level =\< 1.5 x the upper limit of normal (ULN) mg/dL * Aspartate aminotransferase (AST) =\< 2.5 x ULN and alanine aminotransferase (ALT) =\< 2.5 x ULN * Alkaline phosphatase \< 2.5 x ULN * Serum creatinine =\< 1.5 x ULN * Calculated creatinine clearance \>= 30 mL/min using the Cockcroft-Gault formula may be included * Creatinine phosphokinase (CPK) =\< 2.5 x ULN * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L * Platelet count \>= 100,000/mm\^3 (100 x 10\^9/L) * Hemoglobin \>= 9 g/dL * Subject must demonstrate a left ventricular ejection fraction (LVEF) \> 45% by echocardiography (ECHO) or multigated acquisition scan (MUGA) * Male or non-pregnant and non-breast feeding female: * Females of child-bearing potential must agree to use highly effective contraception without interruption from initiation of therapy and while on study medication and have a negative serum pregnancy test (beta - human chorionic gonadotropin \[hCG\]) result at screening and agree to ongoing pregnancy testing during the course of the study, and at the end of study treatment; a highly effective method of contraception is defined as one that results in a low failure rate (that is, \< 1% per year), when used consistently and correctly, such as implants, injectables, combined oral contraceptives, some intrauterine contraceptive devices, sexual abstinence, or a vasectomized partner * Male subjects must practice abstinence or agree to use a condom during sexual contact with a pregnant female or a female of childbearing potential while participating in the study * All ongoing toxicities related to prior therapies must be resolved to grade 1 or better (except alopecia) * Subject must have an Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 or Karnofsky performance scale \>= 70 * Subjects must have one or more measurable lesions, as determined by Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1 assessed by computed tomography (CT) or magnetic resonance imaging (MRI) * Subjects must have a life expectancy of \>= 6 months, as determined by the treating physician * Ability to understand and sign informed consent document * Willingness and ability to comply with the scheduled visits, laboratory tests, and other study procedures Exclusion Criteria: * Known active, uncontrolled, or symptomatic central nervous system (CNS) metastases; a subject with controlled and asymptomatic CNS metastases may participate in this study; as such, the subject must have completed any prior treatment for CNS metastases \>= 28 days (including radiotherapy and/or surgery) prior to the start of treatment in this study and should not be receiving chronic corticosteroid therapy in excess of 10 mg daily prednisone (or equivalent) for CNS metastases; subjects with known CNS metastases must be confirmed radiographically stable by at least one imaging study, at least 28 days from last treatment * Receipt of any type of cytotoxic, biologic, or other systemic anticancer therapy (including investigational) within 2 weeks of enrollment * Prior organ transplantation, including allogeneic stem cell transplantation * Prior treatment with trabectedin * Significant acute or chronic infections including, among others: * Known history of testing positive test for human immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome (AIDS) * Known active infection with hepatitis B or hepatitis C * Active autoimmune disease that might deteriorate when receiving an immunostimulatory agent: * Subjects with diabetes type I, vitiligo, psoriasis, hypo- or hyperthyroid disease not requiring immunosuppressive treatment are eligible * Subjects requiring hormone replacement with corticosteroids are eligible if the steroids are administered only for the purpose of hormonal replacement and at doses =\< 10 mg or 10 mg equivalent prednisone per day * Administration of steroids through a route known to result in a minimal systemic exposure (topical, intranasal, introocular, or inhalation) are acceptable * Known severe hypersensitivity reactions to monoclonal antibodies (grade \>= 3 National Cancer Institute \[NCI\] Common Terminology Criteria for Adverse Events \[CTCAE\] version \[v\] 5.0), any history of anaphylaxis, or uncontrolled asthma (that is, 3 or more features of partially controlled asthma) * Pregnant or lactating females * Known, active alcohol or drug abuse * All other significant diseases (for example, inflammatory bowel disease, uncontrolled asthma), which, in the opinion of the investigator, might impair the subject's tolerance of trial treatment * Any vaccination within 4 weeks of the first dose of avelumab, with the following exceptions: \* Administration of inactivated vaccines, including inactivated flu vaccines, are allowable; however, they should not be given within 2 weeks prior to starting study treatment * Clinically significant cardiovascular disease including cerebral vascular accident/stroke (\< 6 months prior to enrollment), myocardial infarction (\< 6 months prior to enrollment), congestive heart failure with New York Heart Association (NYHA) class II or greater or serious cardiac arrhythmia requiring medication * Severe (requiring active treatment) acute or chronic medical conditions including: colitis, inflammatory bowel disease, pneumonitis, or pulmonary fibrosis * Recent (within the past year) or active suicidal ideation or behavior