I can't imagine there being something readily available that would be heavy enough, large enough, and food-safe—yet printable with a standard printer for use as a 'Chinese' takeout box.
I think that it would have to be at least A3 size for any sort of a usable box—that's 11.7 x 16.5 inches (297 x 420 mm.)
What you need is a simple screen printing set-up to print on pre-made boxes.
A set-up for box printing should be very inexpensive, especially if you're only printing in one color.
Multi-colors would complicate things, but, it is, of course, very doable.