For a brief overview, let's look at some NASA outreach material
The main points being:
- Be a U.S. citizen
- Possess a master's degree* in a STEM field, including engineering, biological science, physical science, computer science or mathematics, from an accredited institution.
- Have at least two years of related professional experience obtained after degree completion or at least 1,000 hours pilot-in-command time on jet aircraft.
- Be able to pass the NASA long-duration flight astronaut physical.
So looking at those one by one:
- Be a U.S. citizen
The US practices birthright citizenship, for people born US territory. Yeager was born in West Virginia, so this checks out.
- Possess a master's degree* in a STEM field, including engineering, biological science, physical science, computer science or mathematics, from an accredited institution.
Yeager graduated from high school, and did not obtain a master's degree (He did however later write a thesis. It's now declassified). However, there's some extra information about this, as noted by the asterisk:
*The master's degree requirement can also be met by:
- Two years (36 semester hours or 54 quarter hours) of work toward a doctoral program in a related science, technology, engineering or math field.
- A completed Doctor of Medicine or Doctor of Osteopathic Medicine degree.
- Completion (or current enrollment that will result in completion by June 2021) of a nationally recognized test pilot school program.
The "nationally recognized test pilot school program" here is the interesting part. Yeager became a test pilot at the Edwards Air Force Base, which is currently the location of the nationally recognized Air Force Test Pilot School. It seems to me that Yeager passes the master's degree requirement by one of the exceptions, being a test pilot. (Several other arguments could also be made, such as being the commandant the USAF Aerospace Research Pilot School)
- Have at least two years of related professional experience obtained after degree completion or at least 1,000 hours pilot-in-command time on jet aircraft.
I'm unfamiliar with how military command works, but it seems Yeager was pilot-in-command for a period of at least 15 years, and this should sum up to considerably more than 1,000 hours of flight time.
- Be able to pass the NASA long-duration flight astronaut physical.
You have waived this already by saying "Had he been willing and physically able to be an astronaut".
As we know Chuck Yeager never became an astronaut. But as a hypothetical scenario, there doesn't appear to have been any formalities stopping him from becoming one.