Local Telecom Expansion Act of 2021 This bill requires the Federal Communications Commission to establish a program for awarding grants to states for the deployment of broadband service in underserved areas by small business broadband providers. A state that receives such a grant must use the funds to make subgrants to small business broadband providers that enable those providers to deploy broadband service in underserved areas in the state. This broadband service must have (1) a download speed of at least 25 megabits per second (Mbps); (2) an upload speed of at least 10 Mbps; and (3) latency that is sufficiently low to allow real-time, interactive applications.