Speeding Therapy Access Today Act of 2021 This bill requires and authorizes various actions to accelerate the development of therapies for rare diseases. The Food and Drug Administration (FDA) shall establish the Intercenter Institute on Rare Diseases and Conditions. The institute shall (1) coordinate engagement with relevant stakeholders, (2) build the FDA's expertise in the review of medical products to treat rare diseases, (3) coordinate regulatory science initiatives related to rare diseases, (4) establish and implement a program to make recommendations to address challenges associated with developing medical products to treat rare diseases in an individual or in very small populations, (5) convene a stakeholder meeting to consider potential amendments to labels for medical products to treat rare diseases, and (6) establish and carry out a program to facilitate voluntary communication between the sponsors of such medical products and third-party payers (e.g., insurance companies). The bill also authorizes the FDA to make grants to assist in developing practices related to the development and production of individualized therapies or therapies to treat very small populations. The bill also establishes an advisory committee to advise the FDA on issues related to the development of therapies to treat rare diseases.