I can't say for sure how many, but of the three farms I've sent mares to over the years (for breeding to a specific stallion at the farm) all three offered the service. Maybe that's because they were all "larger" or at least more well-known farms. Again, I think doing your homework and understanding the "quality" of the farm (their ethics, care for their horses, reputation, etc.) all plays a role.
I am totally what would be considered a small farm, but I have foaled out four mares for other people over the years. Three were mares bred to one of my stallions, one was a mare that just needed a place to foal out. It's very nerve-wracking and unless you've really got the right set-up (cameras, enough eyes to watch, good vet close by) I wouldn't suggest just anyone take on that responsibility. At my new place I don't feel like I have a very good set-up so that is why I am now choosing to send my mare out instead of doing it myself.
I've really come around to the opinion that there are PLENTY of awesome stallions out there, so I'm all for sending a mare or two away (and all the costs/risks associated with it) instead of having to always have my own stallion. Plus, even if a person does have an awesome stallion, you still need outside blood to keep your program moving forward, imo.
And I've heard lots of people say this or that "stallion doesn't stand to outside mares." My experience has been if your mare is nice enough conformationally, easy to handle and you are offering the possibility of a show home should the foal merit it, you will find plenty of awesome stallions that are available.
Good luck in your search! I find searching for the perfect match for a mare to be a lot of fun!