Q. Do I have to buy a boxspring if I get a new daybed?
No. Daybeds are meant to be used as-is, without a spring.
Daybeds are essentially twin beds, put on a decent wood or metal foundation. They do not have to unfold or pull out, so they retain their shape and are thicker than the flimsy mattresses associated with pull-out couches.
You don’t need a box spring, and actually a box spring might mess with the overall functionality of the daybed, in that the daybed is meant to function as a sofa during the day, and a spring might lift the bed too high, making the back of the sofa feel too low.