What insurance company covers OEM parts?

So, I hit a deer, and off to the body shop. My insurance company only wanted to pay for non-OEM parts and I had to pay the difference if I wanted OEM. I lease my car and the terms of the contract require OEM, and I can take a pretty big hit in the trade in if they figure out that I did a repair without manufacturer parts. So… Who does OEM? I’m having trouble finding a company that does. Is this just not a thing?