Inclusion Criteria:  Ability to understand and willingness to provide independent informed consent; legally authorized representative consent and/or power-of-attorney is not allowed. Age at least 18 years at the time of study drug administration Resectable locally advanced rectal cancer (i.e., T3 to T4 or T1-T4 with N1-2 M0). Recommended to receive preoperative radiation therapy Adequate performance status (ECOG of 0 or 1; or KPS of >70). Agree to adhere to lifestyle considerations throughout study duration Agree to not donate blood during the study or for 90 days after the last dose of study treatment.  Exclusion Criteria:  Absolute neutrophil count < 1,500 cells /µL Platelets < 100,000 cells/µL Hemoglobin <9 g/dL Serum creatinine > 1.5 x upper limit of normal (ULN) or calculated creatinine clearance 60mL/min using the Cockcroft-Gault equation Total bilirubin > 1.5 x ULN (>2.0 x ULN in patients with known Gilberts syndrome) or direct bilirubin > 1 x ULN Aspartate aminotransferase and alanine aminotransferase > 2.5 x ULN International normalized ratio (INR) or prothrombin time (PT) >1.5× ULN unless patient is receiving anticoagulant therapy as long as PT or partial thromboplastin (PTT) is within therapeutic range of intended use of anticoagulants. Activated partial thromboplastin time (aPTT) >1.5× ULN unless patient is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants Uncontrolled arterial hypertension, i.e. systolic BP > 140 mmHg, diastolic BP > 90 mmHg. Platelet transfusion ≤ 4 weeks prior to initiating protocol therapy. Presence of any M1 metastatic lesions. Prior pelvic radiotherapy Indication for total neoadjuvant therapy or alternative radiation regimen Active Crohn's disease or another inflammatory bowel disease Any T or N stage disease deemed unresectable by colorectal surgery without neoadjuvant therapy Prior anti-PD-L1 therapy, PARPi therapy, or known germline BRCA-1/2 mutation as patients with germline BRCA-1/2 mutations have an increased risk of severe normal tissue injury to combination radiation and PARP inhibition. Received a live vaccine within 14 days of initiating protocol therapy. Received colony stimulating factors ≤ 4 weeks prior to Day 1 of protocol therapy. Major surgery ≤ 3 weeks prior to Day 1 of protocol therapy (participant must recover from any surgical effects). Investigational therapy ≤ 4 weeks, or within a time interval less than at least 5 half-lives of the investigational agent, whichever is shorter, prior to Day 1 of protocol therapy. Known hypersensitivity to niraparib and dostarlimab components or excipients. Known grade 3 or 4 anemia, neutropenia or thrombocytopenia due to prior chemotherapy that persisted > 4 weeks and was related to the most recent treatment. Known history of myelodysplastic syndrome (MDS) or acute myeloid leukemia (AML) Diagnosis, detection, or treatment of another type of cancer ≤ 2 years prior to initiating protocol therapy (except basal or squamous cell carcinoma of the skin and cervical cancer that has been definitively treated). Known history of ≥ grade 3 immune-related AE with prior immunotherapy, with the exception of non-clinically significant lab abnormalities. Diagnosis of immunodeficiency or has received systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to initiating protocol therapy. Patients with known HIV who have documented detectable viral load or patients with a documented undetectable viral load and a CD 4 count < 350 cells within 6 months of study treatment day 1. Known active hepatitis B (e.g., hepatitis B surface antigen [HBsAg] reactive) or hepatitis C (e.g., hepatitis C virus [HCV] ribonucleic acid [qualitative] is detected). Active autoimmune disease that has required systemic treatment in the past 2 years (i.e., with use of disease-modifying agents, corticosteroids, or immunosuppressive drugs). Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. History of interstitial lung disease. Active or uncontrolled infection necessitating hospitalization or treatment delay. Known serious, uncontrolled medical disorder or nonmalignant systemic disease that preclude eligibility to undergo low anterior resection (LAR). Examples include, but are not limited to, uncontrolled ventricular arrhythmia, recent (within 90 days) myocardial infarction, uncontrolled major seizure disorder, or any psychiatric disorder that prohibits obtaining informed consent Pregnancy. Actively breastfeeding. Declines to use a highly effective method of contraception from screening through 180 days after the last dose of niraparib and after the last dose of dostarlimab.