Testing the Addition of Ivonescimab Combined With Standard Chemotherapy Compared to the Usual Chemotherapy and Immunotherapy Treatment for Patients With Advanced Biliary Tract Cancer

Inclusion Criteria: * Patient must be ≥ 18 years of age * Patient must have an Eastern Cooperative Oncology Group (ECOG) Performance Status of 0-1 * Patient must have histologically or cytologically confirmed adenocarcinoma of the biliary tract including cholangiocarcinoma (intrahepatic or extrahepatic) or gallbladder carcinoma * Patient must not have a diagnosis of ampullary cancer * Patient must have documented metastatic or locally advanced unresectable disease on CT or MR imaging * Patient must have measurable disease as documented on CT or MRI imaging done within 28 days prior to randomization * Patient must not have received prior systemic therapy for current metastatic or locally advanced biliary tract cancer * NOTE: Patients who have previously received adjuvant/neoadjuvant chemotherapy and/or radiotherapy for curative intent non-metastatic disease are eligible if they developed recurrent disease \> 6 months after completion of adjuvant therapy/radiotherapy * Patient must not be on any systemic immunosuppressant therapy other than inhaled steroids, intranasal steroids, topical steroids or systemic steroids up to 10mg prednisone equivalent * Patient must not have received a live attenuated vaccine within 30 days prior to randomization. Patients must also not receive a live attenuated vaccine while on protocol treatment or up to 30 days after the last dose of protocol treatment * Patient must have no contraindication to VEGF inhibitor therapy * Patient must not have significant vascular disease (i.e., aortic aneurysm surgical repair or peripheral arterial thrombosis) within 6 months prior to randomization * Patient must not have inadequately controlled arterial hypertension (systolic blood pressure \> 150 mmHg and/or diastolic blood pressure \[BP\] \> 100 mmHg). Anti-hypertensive therapy to achieve these parameters is allowed * Patient must not have experienced a clinically significant bleeding event within 6 months prior to randomization * Patient must not have experienced gastrointestinal perforation, abdominal fistula, gastrointestinal obstruction, or intraabdominal abscess * Patient must not have had surgery within 30 days prior to randomization * Patient must not be pregnant or breast-feeding due to the potential harm to an unborn fetus and possible risk for adverse events in nursing infants with the treatment regimens being used * All patients of childbearing potential must have a blood test or urine study within 14 days prior to randomization to rule out pregnancy * A patient of childbearing potential is defined as anyone, regardless of whether they have undergone tubal ligation, who meets the following criteria: 1) has achieved menarche at some point, 2) has not undergone a hysterectomy or bilateral oophorectomy; or 3) has not been naturally postmenopausal (amenorrhea following cancer therapy does not rule out childbearing potential) for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months) * Patient must not expect to conceive or father children by using accepted and effective method(s) of contraception or by abstaining from sexual intercourse for the duration of their participation in the study. Patients of childbearing potential must continue contraceptive measures for 6 months after the last dose of protocol treatment, with the exception of cisplatin requiring 14 months for female patients and 11 months for male patients * Patient must not nurse infants for 4 months after the last dose of pembrolizumab, 3 months after the last dose of durvalumab or ivonescimab and for four weeks after the last dose of cisplatin * Patient must have the ability to understand and the willingness to sign a written informed consent document. Patients with impaired decision-making capacity (IDMC) who have a legally authorized representative (LAR) or caregiver and/or family member available will also be considered eligible * Hemoglobin ≥ 9.0 g/dL (must be obtained ≤ 7 days prior to randomization) * Absolute neutrophil count (ANC) ≥ 1,500/mm\^3 (must be obtained ≤ 7 days prior to randomization) * Platelet count ≥ 100,000/mm\^3 (must be obtained ≤ 7 days prior to randomization) * Bilirubin ≤ 2.5 x institutional upper limit of normal (ULN) (must be obtained ≤ 7 days prior to randomization). Patients with Gilbert's syndrome must have a direct bilirubin \< 1.5 mg/dL * Aspartate aminotransferase (AST)(serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT)(serum glutamate pyruvate transaminase \[SGPT\]) ≤ 2.5 × institutional ULN (must be obtained ≤ 7 days prior to randomization). For patients with liver metastases, AST and ALT ≤ 5 x ULN * Creatinine clearance (CrCI) \> 50ml/min or calculated CrCI \> 50ml/min as determined by Cockcroft-Gault (using actual body weight) Cockcroft-Gault Formula (must be obtained ≤ 7 days prior to randomization) * Urine dipstick for proteinuria \< 2+ or 24 hour urine protein \< 1.0 g (within 7 days prior to initiation of study treatment) * Prothrombin time (PT) or international normalized ratio (INR) and partial thromboplastin time (PTT) ≤ 1.5 X ULN (must be obtained ≤ 7 days prior to randomization). This applies only to patients who are not on therapeutic anti-coagulation. Patients receiving therapeutic anti-coagulation are eligible if on stable dose * Patient must not have National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE) Grade \> 2 peripheral neuropathy at the time of randomization * Patient must not have a history of allogeneic organ transplantation * Patient must not have prior documented autoimmune or inflammatory disorders (including inflammatory bowel disease such as colitis or Crohn's disease), systemic lupus erythematosus, Sarcoidosis syndrome, or Wegener syndrome (i.e.: granulomatosis with polyangiitis, Graves' disease, rheumatoid arthritis, hypophysitis, uveitis, etc.). The following are exceptions to this criterion: * Patients with vitiligo or alopecia * Patients with hypothyroidism (i.e.: following Hashimoto syndrome) stable on hormone replacement * Any chronic skin condition that does not require systemic therapy * Patients without an active disease in the last 5 years * Patients with celiac disease controlled by diet alone * Patients with insulin dependent diabetes * Patient must not have uncontrolled intercurrent illness that would limit compliance with study requirement, substantially increase the risk of incurring AEs, or compromise the ability of the patient to give written informed consent * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months prior to randomization are eligible for this trial * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class 2 or better