How do I arrange transportation to Dr. visits?

Depends. Are you wheelchair bound? If so you can call a wheelchair transport company or even contact your public transit to see if they offer a service.

If you are ambulatory, why not use Uber or Lyft? Technically Uber and Lyft also offer wheelchair transport but I've never seen a vehicle that can do that in my area.

Check with your County Office of Aging. They may have special buses.

If you live in a city serviced by a public transportation system....call them. They usually have a service to do exactly what you are asking for.

my experience is that the application process takes quite a while, and the doctor has to sign off on it too.

MM, this topic is raised frequently enough that there are lots of answers and suggestions.   You can check out any of these hits for threads and comments on this topic:

https://www.agingcare.com/search?term=transportation+to+doctor+visits

Senior Centers, private services, and non emergency wheelchair enabled vehicles are good sources to explore.