Livable Communities Act of 2021 This bill creates the Office of Sustainable Communities in the Department of Housing and Urban Development (HUD) to coordinate the provision of long-term affordable housing, transportation, and environmental projects and to administer grant programs established in the bill. The bill creates the Comprehensive Planning Grant Program and the Community Challenge Grant Program to be administered by the office. Grants under the Comprehensive Planning Grant Program must be used for specified activities related to the coordination, development, or updating of a comprehensive regional plan across jurisdictions and agencies. Grants offered under the Community Challenge Grant Program must be used to promote integrated planning and investments across policy and governmental jurisdictions and to implement projects identified in a regional plan. The bill also creates a program that requires HUD to provide loans and loan guarantees to projects that meet specified criteria. Eligible projects must be part of a community-wide development plan, promote sustainable development, and ensure that housing units associated with the project are affordable and initially occupied by lower-income renters or homeowners.