It totally depends on who the character is, we have an awesome shot of my three girls with mickey holding our 2 month old. It made my husband cry, one of the only shots we have of the baby with a character because she was so tiny, and slept the whole vacation. But, there have been other characters that will not, I think it totally depends on who you get at that very moment. They will sit on the floor though, (also depends on where you are at) we have some great shots that way once the baby got a little older.
Don't count on it, but be pleasantly surprised when it happens...