R.D.
Definitely send a card. If she treated her horse like a person, then that is how you should respond to the situation, like she lost a person. It doesn't matter if you know anything about horses or not.
I had to put my horse of 20 years down last year and it still makes me very sad to think about it. I also did everything I could. Not only did I get sympathy cards from friends and family, but the vet that was treating her also sent me one of the most touching cards I've ever seen.
Just remember, treat it like she lost a friend, so no saying, "you can get another", "it was old", or "so what do you do with the body of an animal that big?"