Should we schedule a pre-purchase vet check and what should we have them look at?

We're looking at a horse for my daughter who is 16 and has been riding for 10 years. She shows casually as a hunter jumper and is looking to move up to larger jumps than her pony can handle; he tops out around 2'6". Her trainer is also a trader and just acquired a prospect on the farm that seems promising.

The appendix we're considering is a little bit older at 14. He stands at 17hh which is much more appropriate for my daughters' height. He has a history as a hunter jumper according to USEF and, most importantly, seems to have a great personality. My daughter has ridden him a good bit over the last week and he seems perfectly sound and well-mannered. He is a bit out of shape - apparently his last owner has kept him in a field since November - but he cleared 3" with ease. The kiddo is excited and the horse seems to be happy jumping again.

This is our first time putting down series money for an equine. What should we be looking at medically before moving forward? Thanks!