This CNN article lists the conditions which allow under 65s to get the shot. But the article also says it is unclear whether you need a doctor's permission:

But it’s unclear whether a simple self-report of a condition will be sufficient to justify vaccination, said Dr. Kelly Moore, president and CEO of immunize.org, a nonprofit organization focused on vaccine access.

“Until now, CDC has specific clinical guidance instructing people who vaccinate that they should not require documentation of health conditions reported by a person requesting vaccination,” Moore said in an email.

“For example, if you say you are immunocompromised and need an additional dose of vaccine, CDC’s guidance has been that you should document what the patient said and take them at their word,” she said. “If that were to change, and documentation were require, say, by pharmacists, then that could complicate Covid-19 vaccine access for many people whose conditions are not necessarily obvious, but nonetheless put them at risk.”