Just an honest, open question. I have an idea of like to have for a hat. However, knowing how to go about getting the correct file formats or knowing the cost of what is asked for in advance seems like a foreign language. Much less if there are design corrections back and forth. I’ve dealt with organizational offices doing similar, but never anything private. Is that even a thing?
On comission sites there can be an average price for each artist; you can then add the complexities you want around the art and increase the price based on how difficult it can be in your head. Most of them also list how much art corrections go for, or how many are included.
As for the file format; most hat manufacturers will accept PNG format; but you can also have the PSD in case they don’t so you can convert it easily.
Printing is often done in vector format when possible. So that’d be an EPS or pdf.