Hi, welcome to the crazy world of horses!
What your mare is doing is called "over-reaching". Horses with longer strides tend to catch their front heels with their back feet. Gaited or dressage type horses tend to do it most, though any breed could do it. High speed performance horses also do it occasionally.Sometimes they actually pull off their own shoes by doing it. Or your farrier could be causing the problem.
First, yes, something called bell boots can help protect her front feet from bruising, at least while you ride, which is usually when a horse is most apt to do it. I dont' advise leaving them on 24/7 because they can chafe and rub and hold in moisture and cause their own little problems.
Your farrier is probably trimming her different to try to fix the problem. Most will leave the hind feet a little longer in the toe to lengthen what's called "breakover" and keep the foot on the ground a little longer to give the front foot time to get out of the way. Then they tend to shorten the toe in the front foot to speed it up. The problem with this is that it doesn't always work, plus, it messes with the tendons in the legs and can set the front feet up for other long-term soundness issues.
This is certainly not an emergency and you would be better off talking to the farrier than the vet in this case. You might even get a second opinion from another farrier/trimmer to see if the problem is the mare's build or the current farrier's work. She may always be prone to catching herself, so again, bell boots are a blessing for rides in that instance. I'm a barefoot trimmer, not a farrier, meaning I have the same job, but I solve problems without the use of metal shoes. I want the hoof to be as natural as possible, so I would probably see things a little differently than a traditional farrier, but a second opinion, regardless would be worth the time. You may still prefer your current guy, but a second opinion is helpful sometimes.