This is one of the common behaviors of toddlers.
Toddlers may not want to eat food that their parents give to them. But when their parents take them to a group of other kids or people who are eating food.
These toddlers may stare at the food & their eyes express as if they want others to give them some to eat.
These toddlers may or may not show their hands towards them. So, I think the verb "begging" may not be suitable.
I found this word
sponge informal + disapproving : to get money, food, etc., from (someone) without doing or paying anything in return
She always sponges meals from us.
[no object] — often + off
She lives at home and sponges off her parents.
How to express this toddler behavior: "She always stares and seems to beg for food of other kids" or "She always sponges food from other kids"?