I don't think its sold anymore, but I used to use a yi lite to record my games. Not the best quality but it did the job and was about $60. If you search action camera on amazon, there's quite a few options under $100, but I couldn't say if they're any good or not.
I recently picked up a gopro hero 9, it was $350 but the quality is much better. You might be able to find an older one elsewhere for ~$100.
When I play, I have a gopro suction cup attachment I use that I put on the inside of the glass behind me. I've recorded my games for about 3 years and it hasn't got hit yet!
I personally have a Hero3 I won five years ago in a video contest, Crosstour Action Camera I use for my rear on my commuter (when I remember to charge it). And for my race bike I have my Fly6(ce), but I will use it on my commuter if I forget to charge my other rear camera.
Here is a list of common ones:
Crosstour Action Camera amazon link
Yi Lite amazon link
Contour ROAM2 Waterproof Video Camera amazon link
Yi 4K Action Camera amazon link
Garmin Virb Elite amazon link
Sony HDRAS100V/W amazon link
GoPro Session amazon link
Fly12(ce) camera and front light amazon link
Xiaomi Yi 4K Action Camera 2 NIS Store link
Fly6(ce) Amazon link