Leflunomide for the Treatment of Severe COVID-19 in Patients With a Concurrent Malignancy

Inclusion Criteria: * Documented informed consent of the participant and/or legally authorized representative. Assent, when appropriate, will be obtained per institutional guidelines. Cognitively impaired subjects may enroll in the phase 2 portion if adequate psychosocial support is provided * SARS-CoV-2 infection confirmed by a PCR-based test within 4 days prior to enrollment * COVID-19 disease baseline severity of Severe according to FDA guidance, as defined by: * Symptoms suggestive of severe systemic illness with COVID-19, which could include any symptom of fever, cough, sore throat, malaise, headache, muscle pain, gastrointestinal symptoms, or shortness of breath at rest, or respiratory distress * Clinical signs indicative of severe systemic illness with COVID-19, such as respiratory rate \>= 30 per minute, heart rate \>= 125 per minute, SpO2 =\< 93% on room air at sea level or partial pressure of oxygen (PaO2)/the fraction of inspired oxygen (FiO2) \< 300 * Active cancer requiring systemic treatment within the last 2 years. Subjects should not have received the following therapies for their malignancy within the indicated time frames: * Local radiation therapy within 2 weeks prior to enrollment. If the involved field is small (single nodal area), 7 days prior to enrollment is allowed * Chemotherapy within 2 weeks prior to enrollment * Major surgery within 2 weeks prior to treatment * Autologous hematopoietic stem cell infusion within 12 weeks prior to enrollment * Antibody therapy, chimeric antigen receptor (CAR) T cells, or other biologic therapies within 12 weeks prior to enrollment * Allogeneic hematopoietic stem cell infusion within 16 weeks prior to enrollment These time frames should be considered the minimum allowed interval and may be longer per the judgment of the investigator * Adverse events related to prior cancer therapy must have recovered to =\< grade 1 or to baseline * Subjects must be able to forgo systemic cancer therapy for \~39 days (14 days treatment/placebo + 14 days monitoring + \~ 11 days cholestyramine) * Absolute neutrophil cunt (ANC) \>= 500/mm\^3 (to be performed within 4 days prior to day 1 of protocol therapy unless otherwise stated) * Platelets \>= 25,000/mm\^3 (to be performed within 4 days prior to day 1 of protocol therapy unless otherwise stated) * Total bilirubin =\< 1.5 x upper limit of normal (ULN) (unless has Gilbert's disease) (to be performed within 4 days prior to day 1 of protocol therapy unless otherwise stated) * Aspartate aminotransferase (AST) =\< 2 x ULN (to be performed within 4 days prior to day 1 of protocol therapy unless otherwise stated) * Alanine aminotransferase (ALT) =\< 2 x ULN (to be performed within 4 days prior to day 1 of protocol therapy unless otherwise stated) * Creatinine clearance of \>= 60 mL/min per 24-hour urine test or the Cockcroft-Gault formula (to be performed within 4 days prior to day 1 of protocol therapy unless otherwise stated) * Agreement by females and males of childbearing potential\* to use an effective method of birth control or abstain from heterosexual activity for the course of the study until teriflunomide levels are verified to be less than 0.02 mg/L (0.02 ug/mL) for patients given leflunomide, or until unblinding occurs for those given placebo. Contraception should also be used for the duration of administration of SOC drugs during this study for the duration recommended in the prescribing information. * Childbearing potential is defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only) Exclusion Criteria: * Evidence of acute respiratory distress syndrome (ARDS), defined by at least one of the following: endotracheal intubation and mechanical ventilation, oxygen delivered by high-flow nasal cannula (heated, humidified, oxygen delivered via reinforced nasal cannula at flow rates \> 20 L/min with fraction of delivered oxygen \>= 0.5), noninvasive positive pressure ventilation, extracorporeal membrane oxygenation (ECMO), or clinical diagnosis of respiratory failure (i.e., clinical need for one of the preceding therapies, but preceding therapies not able to be administered in setting of resource limitation) * Shock (defined by systolic blood pressure \< 90 mm Hg, or diastolic blood pressure \< 60 mm Hg or requiring vasopressors) * Evidence of multi-organ dysfunction/failure * Pre-existing acute or chronic liver disease * Patients with indolent local malignancies or pre-malignant conditions including but not limited to: * Smoldering multiple myeloma or monoclonal gammopathy of undetermined significance (MGUS) * Basal or squamous cell carcinoma of the skin * Carcinoma in situ of the cervix or breast * Incidental histologic finding of prostate cancer (T1a or T1b using the tumor node metastasis \[TNM\] clinical staging system) or prostate cancer that is curative * History of allergic reactions attributed to compounds of similar chemical or biologic composition to study agent * Secondary bacterial, fungal, or viral infections that are not adequately controlled * Any other condition that would, in the Investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures * If human immunodeficiency virus (HIV)-positive: CD4+ T cell count \< 200 * Positive for tuberculosis antigen (e.g., T-spot test) * Presence of liver metastasis * Gastrointestinal (GI) malignancies associated with malabsorption and inability to take cholestyramine * Steroids, except for low-dose replacement or high-dose for management of acute symptoms such as ARDS * Any new immunosuppressive medication in the 4 weeks prior to enrollment, excepting agents used for treatment of COVID-19 that may also have immunosuppressive properties * Medications that are CYP1A2 inducers, CYP2C8 inhibitors, and vitamin K antagonists, if these medications are not approved by the investigator. CYP1A2 inducers, CYP2C8 inhibitors, and vitamin K antagonists that are approved by the investigator are allowed * Concurrent administration of live vaccines * Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)