My son can not have dairy and they are always very good about providing a meal that he can eat as well. I am sure you will have no problems getting a vegetarian meal at LTT or anywhere else on Disney property. When making your reservation I would have it noted on there so that they are aware ahead of time and can accomodate you better. Disney is the only place that we can go and get a dairy free meal for our son without any hassle. They are great dealing with special needs and or requests.
Enjoy your trip.