You should place the syringe all the way toward the back of the mouth and little into the throat, she may do what you consider a gagging motion but she won't throw up because of it. The problem is if you just put it in her mouth she could possibly aspirate because of poor placement because their airway is under their tongue. However I don't know anything about doxycycline specifically, I want to say it's the same med as Baytril which could very well be the problem. Almost all chameleons don't do well Baytril , however I haven't heard of many who have a reaction as violent as hers. However Baytril is known for causing dehydration, disinterest in food, and eventually death. Hold on lemme look it up and make sure.
Edit: I was right, doxycycline is another name for Baytril, which I strongly recommend asking for a different antibiotic. Not just because of her reaction, though that is one of the reasons, but because in general chameleons so not react well to it as a whole.