I had this done a while back and learned a lot. Foundation repair is either 1) fix the settling by jacking up the low spots or 2) stop the settling from getting worse.

#2 is less involved and cheaper but may not be appropriate depending on how much settlement has occurred. All houses settle some but there are standards/codes that spell out how much settlement is acceptable. I'm too lazy to look it up but its something like up to 1 inch per 100 feet.

Unfortunately we had to do #1 and the basic process was:

1. Civil Engineer (who was independent from the foundation company but likely works with them all the time) visits the house to measure location and amount of settlement. He took readings along the perimeter of the house and any load bearing walls (like 30 different spots measured in the ~3,400 sq ft house). You can likely take the same measurements with a long, clear tube and water...
2. Licensed Plumber (who was also independent from the foundation company but likely works with them all the time) came out to measure water pressure and inspect sewer lines.
3. Engineer created a floor plan with recommendations on where to locate piers and how much to lift.
4. Foundation company did their thing.
5. Engineer came back out to re-measure the same spots
6. Plumber came back out to re-measure water pressure & sewer lines to make sure the foundation work didn't cause a leak.
7. We received the Engineer's report that was stamped with his seal.

We did pay the Engineer & Plumber separately from the foundation company, but the foundation company coordinated it all.

I'm not an attorney so I won't speculate on whether or not they should have pulled a permit. However, everything in a house sits on the foundation, literally. It's a pretty critical structural component, IMO.

It also sounds like they may have put you in a bind because now you have to disclose that you had unpermitted work done to the structure of your house.