My trailer wiring was not working correctly. So I took it to a shop. The shop told me my left turn and trailer brakes will not work. Said the problem was the integrated power module needed to be replaced and would cost me $900. Is this why my trailer wiring will not work? Can I fix this myself?

It could be since the factory trailer connection goes through the IPM (fancy fuse block).
What I would do is shop for an aftermarket trailer connector that can be spliced into the rear harness.

Agreed. Shop around for a less than brand new one not from the factory.

2003 dodge ram integrated power module | eBay

Seems you do have cheaper options, and it may be a good Saturday afternoon project for you.