Treat cereal boxes the same way you would glossy paper. We try not to use it for the same reasons.

My worms eat cereal boxes just as quickly as cardboard. I also feed them all the glossy paper that comes into the house, and over the years, I've not seen any problems. I believe most glossy paper just has a coating of clay on the surface.
Renais

A coating of clay on the paper, and often a coating of varnish over the ink. The paperboard itself is fine.

Whenever I get bored, I peel the printed layer before shredding.

It's easy to peel off the thin glossy coloured layer after wetting the cereal box, crumpling it into a ball, squeezing all the water out and wetting it again.

Cereal boxes are pretty cheap material , I stay away from that, but will use corrugated with all tape and labels removed. If you dont remove them you will find it in the finished harvest mix not composted. Sometimes it may be easier to just remove that portion of cardboard.

Often nowadays cardboard mailing boxes a la Amazon boxes are using brown paper tape, not plastic tape, but I haven't noticed whether that tape breaks down completely in the bin. I have so much other stuff available to use for worm bedding that I mostly put my online shopping boxes out for regular recycling.

If you spoil your worms, they will be spoilt worms. Whatever they can't eat will remain. Avoid citrus primarily. The rest is small stuff. :)