WDW is wonderful place for vegetarians in my opinion. I have had so many wonderful meals and I sometimes (okay, every time) dread coming back home!
There are plenty of CS places that serve veggie burgers. My favorite CS in the MK is Cosmic Ray's. They serve tasty veggie burgers with a toppings bar that I just love! They also have a veggie wrap that I am looking forward to possibly trying this upcoming trip! Columbia Harbour House also has a veggie sandwich that has hummus.
Epcot has several options for CS as well. Sunshine Seasons is my favorite CS at Epcot for vegetarian variety. They offer vegetarian sandwiches, vegetarian asian offerings and salads. Liberty Inn also offers veggie burgers and they are supposed to offer a veggie wrap, but I personally haven't tried it.
As far as the AK goes up until Tusker House became a buffet that was my go to lunch restaurant for CS. I have also been to Restaurantosaurus for a veggie burger. I do know that Flame Tree BBQ has a fruit plate that sounds really tasty, but I personally have not been there. The seating area for FTB is outdoors and I've heard that the birds can be aggressive. I'm terrified of birds, so I have yet to try the food there.
I can't offer you any suggestions for your brother or sister-in-law for HS. I'm sure someone else can fill you in here.
Every TS offers at least one vegetarian entree. And I have never had any troubles finding food at buffets. I'll go ahead and mention some of my favorite TS for you.
AKL - Boma
Tasty, tasty buffet with more than enough for me to eat! I love this place and consider it a must-do.
GF - 1900 Park Fare
I love the buffet here, it's one of my favorites. The strawberry soup is heavenly!

My husband likes to joke that I will have three bowls of strawberry soup and for dessert I'd have more strawberry soup.
Poly - 'Ohana
While it is a meat lovers paradise, I've had great vegetarian offerings here. I have read amazing reviews with Chef TJ is working, but I have yet to try any of his food. But I'm always more than satisfied with what I get. Plus they have an amazing dessert..
MK - Liberty Tree Tavern
This is my family's favorite TS meal at the MK. They have wonderful vegetarian offerings in my opinion. And if we happen to go for dinner, I could make a meal just on the mashed potatoes, rolls and salad alone.
Epcot - Marrakesh
They have a wonderful veggie couscous. Lots of vegetables and very filling. Only reason we aren't going this trip is because I'm pregnant and certain smells are setting off my stomach.

I'm going to miss the couscous.
Epcot - Biergarten
Yep, it's a German buffet, but boy oh boy are the sides tasty! Plus I love the entertainment! I have never left this place hungry!
I hope this helps! There are quite a few vegetarians and vegans out there! Check out the reviews section for more help!
