My 1999 jeep grand cherokee blower motor stops blowing when I accelerate.

Well, I have to admit that's the first time I've heard of that happening. Usually when the blower motor becomes intermittent like that, it's due to bad brushes in the blower motor itself, or the blower motor resistor. Typically, when you lose a speed or two, it's the resistor. I would lean more towards your blower motor going bad. The next time it happens, have your wife or buddy smack the blower motor with their hand or a long ratchet. See if it kicks back on. That will only work if it's located under the glove box which I think it is. If it comes back to life, it's your blower motor that needs replacement.