Do widgets support input from users (not admin users)

I am trying to create an autocomplete widget for wordpress to add to an existing custom theme.

My question is if this is even possible or are widgets strictly only for displaying information?

The only way I see to accept input is from the admin page in the function update() and not from the front-end.

If this is not possible then what would be the best way of doing this without manually changing the theme a lot?