Agreed with others that suggested finding a local person/shop. Maybe check Yelp for embroidery. It is a great idea. We have done jackets, pads, stall guards and even folding chairs! Just realize that unless you go with a horse-specific shop, you will want to be very specific about sizes and locations of the embroidery. They won’t necessarily know how the pad goes on the horse!
I got lucky - a neighbor does machine embroidery out of her home.
“it’s now illegal for AAs to show with a barn logo on their pad. Really annoying, since we used to love competing in our “team” pads.” I’m not sure that is the case - USEF rules: “Competition award pads and stable name pads are permitted.”