Hello!
Most people these days will use a weight/height tape. THey put the tape around the girth/heart girth of the horse and calculate it's weight. Or they could use a forklift and a sling to carry the horse while a weight is attacked. Much like a fishing weight. However, the tape measure is used most often.
I hope this helped!
I am, yours most sincerely,
SuperHeleprThingy