I can find no mention of a TPMS requirement in my state. My older car has a malfunction in its TPMS and has for 10+ years. The repair cost I was quoted was ridiculous and I just periodically check the tire pressure with a normal gauge.

This vehicle has passed all required inspections for that entire time and the TPMS system hasn't even been mentioned in that context.

edit: I found the claim that TPMS was required interesting and a quick web search shows that 13 states in the US don't require vehicle safety inspections AT ALL.