From https://metaculus.com/questions/17110/us-or-ca-license-to-train-ai-before-2026/

As of May 2023, US law does not require any license or permit before training large AI models. One way US policymakers may address risks from AI is to require those who train large models to first apply for a license, as Sam Altman, CEO of OpenAI, called for in DC on May 16, 2023. Reuters also reported an OpenAI staffer calling for a licensing agency for AI.

Many similar regimes exist in US law - for example, licenses to drive cars or fly planes, or permits to construct buildings in various places. Many are at the state level - since most all Frontier AI labs are headquartered in California, and California generally regulates these things more than other states, this question considers state licensing as well.

A list of large AI training runs and who did them can be found here, this could be a rough list of who would require a license.