Good morning,
I am a long arm quilter and I also do some machine embroidery so I think I know what happened to your piece. They did not put stabilizer (or used wrong kind of stabilizer) behind the material and they did not pull the piece straight and smooth in the hoop, that's what caused the wrinkles.
If you pick out the stitches, which is nearly impossible without damaging the fabric because they are so tightly stitched, it will definitely show on the fabric.
I think the place you took it to is used to doing ball caps and other items of that type and not cotton fabrics for quilts, and you use different types of stabilizer for the different items.
My suggested solution, after asking for my money back, is to do some stippling, or even micro stippling in the area around the letters, this may calm the wrinkles and puckers into submission.
Such a cute piece, it's a shame it wasn't embrodered correctly.
Longarm