Q:

In National Velvet why didn't they use a stunt double for Elizabeth Taylor?

In National Velvet [1944] Elizabeth Taylor, as a young actress, fell from a horse and broke her back; this caused her a lot of pain for many years. 
Surely a stunt double could have been found. There are small people in the world [e.g in Game of Thrones] or camera work could have made someone look her size. For someone young and inexperienced, that was a huge risk from the studio. 

A:

They did
Alice van Springsteen was the stunt double for Ms Taylor in National Velvet

Her first work as a film stuntwoman came in Will Rogers' last movie, "In Old Kentucky" (1935). A member of the Cowgirl Hall of Fame, she went on to serve as a stunt double for Elizabeth Taylor (in 1944's "National Velvet"), Marian Davis, Jane Wyman, Ingrid Bergman and Barbara Stanwyck (in "The Big Valley" TV series of the 1960s.)
Hollywood Reporter

I'm still looking for a full report of the accident but there are cases where it's necessary for the actual actor to perform minor stunts. There may be minimal risk but that does not mean there is no risk.
A horse is a large creature and you're a long way off the ground....falling off hurts!

It also seems that "Pie" (the horse) was not the gentle creature one might imagine..

The Pie was a 7-year-old thoroughbred named King Charles, a descendant of famed racehorse Man O War. He was owned by a society woman at the Rivera Country Club in Los Angeles who had trained him as a show jumper, and she offered to let Elizabeth Taylor ride him while at the country club. Elizabeth immediately fell in love with the horse, and he was acquired by MGM for $800 to star in National Velvet with her. She undertook a rigorous training routine to prepare for the movie, and rode King Charles for 90 minutes every day while also feeding and bonding with him at the Rivera Country Club for the rest of the day.
From the beginning, King Charles was a difficult horse to get along with. He wouldn’t listen to commands, and he regularly bit crew members, once seriously injuring a trainer who was trying to make him play dead for a scene. The only person he would listen to was Elizabeth. The two had developed a special bond, which is very evident in the final film.
Source

However, this incident just appears to have been an unfortunate accident.

