What’s this transmission part called?

It's called an inspection cover. It also serves as a drain.

I have bought one at the Ford dealer before.

Could also try your local salvage yard. Note this part would be in any 4R70W transmission. It does not have to come from a Mustang.
 
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In a "pinch" you could out in a freeze plug you should be able to get from your local auto dealer. you will have to replace it again if you ever remove it, but I believe they are cheap, as far as the size is you would just have to see what they have.
Good Luck