Body shop that does rust mitigation
 

I have a 1995 Miata that has a rusty spot behind the drivers door.

I have been to 4 body shops on 441, and all of them say they do not deal with rust.

Anyone know of a shop that can at least stop the rust from spreading?

Having built (OK rebuilt) 8 early Mustangs from the ground up I know what you are talking about. If this is surface rust and NOT an actual hole or real deep) you can buy at any auto parts store (1) rust converter, (2) zinc primer, and if necessary, (3) primer sealer.
If there is real deep rust (most of the metal is gone, or an actual hole, then you are into having to weld in replacement metal. Sometimes you can use a fiberglass patch, I have done both. Of course the final step is getting matching paint and the final finish.

I believe the problem you are running into is that rust is VERY hard to stop. Go to the Eastwood company website they have excellent info on this subject and I have used many of their products with excellent results.

There is a classic/antique car restoration company that I have seen advertised, if I can find or remember the name I will post another reply.

"Stop the rust from spreading," I am not familiar with your car and I can't see how bad the rust is. Car body rust is the result of water not draining properly. Often over time the drainage holes plug with dirt or they are covered with undercoating. Look up where the holes are and clean them.

Dealing with the rust? You are only seeing what has rusted through. It is likely it also exists elsewhere. Driving on salted roads? Down the beach? As a kid my uncle bought a car that had been parked on the beach an Olds 98 with a huge engine. I think he paid like $100 for it. If, you looked down the floor was so rusted you could see the road below you.

Replacing panels is cheap, easy, and profitable. Cutting out rust, welding patches, and shaping panels is far more work.

Let us know what shop ends up doing your work and how it turns out.

Try Imagination Engineering in Apopka. They restore rusted out classic cars.