What are these broken trim pieces called and where can I find them?

Those appear to be the remnants of the stock mudflaps. You can buy OE replacements online or at your local dealer, or you can buy aftermarket flaps that are a little more forgiving as the stock ones are chunky and rigid.

 

What size tires are you running? Chances are the flaps were trimmed due to rubbing issues. I had the same problem... ended up trimming the oem flaps and molding with heat gun to fix.

 

Is your first picture the leading edge of the wheel well? I don't have anything there.

The trailing edge - my stock mud flap:

 

Interesting. Yes, that looks a lot like what I’m seeing, and for sure my mudflaps have all been removed by the prior owner. But do the front mudflaps have two components to them? Feeling around, there nothing jagged or cut about them. It is just looks like a secondary component that is cracked on both sides.

 

Interesting! Mine is stock for sure, I wonder if they changed (omitted) that front edge trim over the years?

 

I guess I could just go to the dealer and ask them. I was planning to order an OEM bed mat anyway. Thanks for looking!

 

This shows what the whole mud flap looks like when mounted. Not quite sure why the previous owner trimmed it like that.... easier to just remove it all together and plug the holes.
https://www.youtube.com/watch?v=u94VTZGuKMo

 

Why not just leave it as is?