Hi Cez! Sounds like you're planning a fun birthday trip! Although the appearance of a particular princess cannot be guaranteed at Akershus, Ariel often greets guests there at breakfast, lunch and dinner. If you want to dine at Akershus, but the items on the menu don't meet your dietary requirements, I recommend that you contact Disney's
Special Diets department by calling 407-824-5967 or emailing SpecialDiets@DisneyWorld.com. If you give them the information about your allergies and tell them you're a vegetarian, they can contact the restaurant to make sure that the chef can accommodate your needs.
Ariel has often been spotted at breakfast, lunch, and dinner at Cinderella's Royal Table too, so you may want to review the menu offerings there to see if they're more compatible with your allergies and vegetarianism.
If you happen to dine at a character meal and Ariel isn't appearing that day, look for her in Adventureland in the Magic Kingdom. She usually appears there daily -- check the Times Guide when you enter the park for her schedule.
Hope you have a magical 24th birthday!