Ohana is now an excellent choice for vegetarians! er, I should say, as long as they are tolerant of others at the table eating meat.....LOTS and LOTS of meat. It's kind of a 'meat-fest'. IF yor firend eats shrimp though, she'll find PLENTY of food among the regular offerings.
You could tell Disney Dining when you make the reservation that you have a vegetarian. I think they only request 24 or 48 hrs noticve, and actually - most WDW restaurants can prep veggie stuff onthe fly, but it's a helpful courtesy. Disney is generally better able to serve your needs if you inform them in advance.
TJ is the best chef on WDW property! I first met him quite a few years ago when he worked at Boma. If you get to meet him, it will most likely be one of the highlights of your visit to WDW! I can't say enough good things about him.
Generally, WDW TS dining is a great place for vegetarians, esp if you let them know in advance.
Again I will say, however, there is MUCH food at 'Ohana. Much of it meat. If you aren't a really big eater, and you don't like to throw away large amounts of food, it's a good idea to ask for half portions of all the sides. Even if your veggie friend says nothing to the staff, there are a number of good sides she can choose from: salad, steamed broccoli with sugar snap peas (I think, usually - maybe some other veggie with the broccoli), lo mein noodles. Those alone are enough for a meal - add shrimp and she'll be stuffed!
Last, don't forget dessert. Just when you are stuffed to the brim, they bring yummy bread pudding with banana's foster sauce (like caramel - very tasty) and ice cream on top. The ice cream is just so-so, but the bread pudding is very good.
Beyond that, most restraurants at WDw have a vegetarian option on their regular menu, and amost have a fish option on the menu (esp if your friend eats salmon). If your firend doesn't find anything she wants on the regular QS menu at any given place, she may be able to order a kids' meal. Most TS places can also make an adult sized portion of any kid's meal - but you won't get a kid's menu unless there is a kid (aged 3-9) in your party. Just about ALL the kid's meals come with real portions of veggies (though at some places the adult vegetable listed is more of a garnish.)
Another place you may want to try is The Wave.