Palbociclib and Sasanlimab for the Treatment of Advanced Clear Cell Renal Cell Carcinoma (ccRCC) or Papillary Renal Cell Carcinoma (pRCC)

* INCLUSION CRITERIA: * Cytologically or histologically confirmed clear cell renal cell carcinoma (presence of a clear cell component) (ccRCC) (Cohort 1) or papillary renal cell carcinoma (pRCC) (presence of a papillary component) (Cohort 2) * Participants must have advanced RCC with at least one measurable lesion as outlined in RECIST 1.1. * Participants with ccRCC (Cohort 1) must have received checkpoint inhibitor therapy and must have received or been ineligible to receive a VEGF pathway antagonist (as a single agent or as part of a combination) * Participants with pRCC (Cohort 2) can be treatment-na(SqrRoot) ve or have previously received systemic treatment for pRCC * Age \>= 18 years * ECOG performance status \<= 1 * Adequate hematologic function at screening, as follows: * Absolute neutrophil count (ANC) \>= 1,000/microliter * Hemoglobin (Hb) \>= 9 g/dL with no blood transfusion within 2 weeks prior to treatment initiation * Platelets \>= 100,000/microliter * Adequate renal and hepatic function at screening, as follows: * Serum creatinine \<= 1.5 x upper limit of normal (ULN) OR, if \>1.5x ULN, creatinine clearance (CrCl) \>= 30 mL/min/1.73 m\^2 (calculated CrCl (CKD-EPI or calculated eGFR provided by laboratory)) * Total bilirubin \<= 1.5 x ULN OR in participants with known or suspected Gilbert's syndrome, total bilirubin \<= 3.0 x ULN * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \<= 2.5 x ULN, (unless liver metastases are present, then values must be \<= 5 x ULN) * Participants serologically positive for hepatitis C virus (HCV) are eligible if HCV viral load is undetectable * Participants serologically positive for human immunodeficiency virus (HIV) are eligible if they are on stable antiretroviral therapy for at least 4 weeks before treatment initiation, have no reported opportunistic infections or Castleman s disease within 12 months prior to treatment initiation, have a viral load that is undetectable by quantitative polymerase chain reaction (PCR) and CD4 count \>= 200 cells per cubic millimeter * Participants with brain metastasis are eligible if at least 4 weeks status post radiotherapy or surgery before treatment initiation with no evidence of progression or associated symptoms * Women of child-bearing potential (WOCBP) must agree to use one (1) highly effective method of contraception (e.g.,hormonal, intrauterine device (IUD), surgical sterilization) prior to study entry, for the duration of study therapy, and for up to 6 months following the last dose of any study agent(s). Women must refrain from donating eggs during this same period. NOTE: WOCBP is defined as any female who has experienced menarche and who has not undergone successful surgical sterilization or who is not postmenopausal. * Men with female partners of reproductive potential and pregnant partners are required to use a condom (even after vasectomy), during treatment and for at least 6 months after the final dose and must refrain from donating sperm during this same period. * Breastfeeding participants must be willing to discontinue breastfeeding from study enrollment through 6 months after study treatment discontinuation * Participants must be able to understand and be willing to sign a written informed consent document EXCLUSION CRITERIA: * Prior treatment for RCC with chemotherapy, hormonal therapy, immunotherapy, treatment with an experimental agent, and/or radiation therapy within 4 weeks or 5 halflives, whichever is shorter, prior to treatment initiation * More than four prior lines of systemic therapy in the metastatic setting * Participants who have wound dehiscence from prior surgeries * Active inflammatory bowel disease, chronic diarrhea, gastrointestinal malabsorption, gastrointestinal anastomosis, or any other condition that might affect the absorption of palbociclib * History of allergic reactions attributed to compounds of similar chemical or biologic composition to the study agents * Prior history of grade \>=3 immune-related adverse event(s) with checkpoint inhibitor therapy. Note: participants who had endocrine toxicity of grades 3 or 4 are eligible * An active autoimmune disease. Note: participants with type 1 diabetes, eczema, vitiligo, alopecia, psoriasis, hypo- or hyperthyroid disease, adrenal insufficiency on systemic oral corticosteroid therapy (\<= the equivalent of prednisone 10 mg/day) or other mild autoimmune disorders not requiring immunosuppressive treatment are eligible. * Participants receiving systemic corticosteroids at doses equivalent \> 10 mg/daily of prednisone, cyclophosphamide, azathioprine, methotrexate, mycophenolate mofetil, sirolimus, thalidomide, or anti-tumor necrosis factor \[anti-TNF\] agents. Note: participants on steroids through a route known to result in minimal systemic exposure (topical, intranasal, intro-ocular, or inhalation) are eligible * Prior allogeneic/autologous bone marrow or solid organ transplant * Participants with current or past hepatitis B (HBV) infection * Participants with a history of interstitial lung disease, non-infectious pneumonitis, or untreated active/latent pulmonary tuberculosis (TB) * Participants taking medications that are strong inhibitors or inducers of CYP3A (https://www.fda.gov/drugs/drug-interactions-labeling/drug-development-and-druginteractions-table-substrates-inhibitors-and-inducers#table3-2) within 21 days or 5 half-lives of the agent (whichever is shorter) prior to initiation of study therapy * Participants taking any herbal supplements within 14 days prior to initiation of study therapy * History of a non RCC malignancy within 2 years of treatment initiation except for the following: adequately treated localized skin cancer, ductal carcinoma in situ, cervical carcinoma in situ, superficial bladder cancer, or other malignancy which does not require treatment at the current time per Standard of Care * Pregnant women (confirmed by beta-HCG serum pregnancy test performed at screening) * Uncontrolled intercurrent illness that would limit compliance with study requirements, evaluated by history, physical exam, and chemistry panel.