Hi, I’m writing a proposal letter to a big publishing company asking if I can use a short story they published. I’ve never really done this before and could use some help. And of course, any tips you can give me would be very helpful!

This wouldn't be a proposal letter. It would be a simple letter asking for permission. Here are two variations that you can combine or adapt in some way for your own purposes --

Sample letter requesting permission to use copyrighted material without a charge

Letter granting permission use copyrighted material