If she's at a Regency ball there is a good chance she doesn't pick up something from the table and eat it. It would depend to some extent on the circumstances, but in most cases I believe she would have to wait for a gentleman to offer to bring her refreshments, and then she would eat whatever he decided to bring her. If he was a true gentleman he might choose carefully and perhaps come back with a plate containing a variety of things, in the hopes that something there would suit her. If she was unlucky, he would be one of those fellows who think they know what ladies want (because all women want the same thing, right?) or at any rate what they ought to have.
Are you thinking something that would be finger-food (like biscuits, or slices of bread and butter, for instance) or something she would eat with a fork?
At tea time, I don't believe people served themselves. The hostess, or a female family member designated by her, would pour, and someone else would pass around. Servants, or family members, might pass around sandwiches, etc.
Tea might be served at a ball, although I doubt it, because dancing is hot work and ballrooms are overheated. If it was, it would be another one of those refreshments someone would probably have to fetch for her.
I imagine that wallflowers spent a lot of time hungry and thirsty.