My 2006 Hummer H3 is showing SERVICE TPM.

I think you mean the TPMS. Every tire has a sensor that sends a signal to the computer telling it tire pressures, if the pressure is too low it turns on the light. As this is a 06 it should already be on at least its second set of sensors, on some models you can access the main display menu and see just what sensor is showing failure, if this model doesn’t have that feature then most tire shops have the equipment to do a scan to be sure, this old it could be a wiring issue. If it is sensors they may have them in stock or can get them overnight. You can also purchase them yourself from Amazon, EBay and some parts stores. I recommend that you purchase an entire set and replace them all at the same time.