I want to buy and install an amplifier for a new sound system for a car. I'd like the amplifier to not clip, and I'm willing to sacrifice a little bit of dynamics for extra loudness when I want it.

In the world of music production we regularly obtain these goals with an effect called a brickwall limiter. It's basically a special type of compressor that guarantees that the sound never passes a certain threshold.

Do they make car audio amplifiers with brickwall limiters (or at least compressors) inside them?

So you want to compress the crap out of your peaks to bring up your average level?  Not at all the right use of a BW limiter.  If you want more loudness then scale your system appropriately.  It will cost you less and sound better.