Going from the information from a plastics compatibility chart I found online, a number of common plastics have excellent durability against sodium hydroxide (aka soda lye), such as polyethylene (PE) and polypropylene (PP). When I restored my cast iron pan a few months ago, I used a lye-based oven cleaner sprayed into the pan and placed it into a plastic garbage bag made of low-density polyethylene (LDPE). The bag itself did not dissolve.

That said, my understanding of plastics compatibility is that test is whether a chemical will not degrade the useful properties of the plastic. So a non-compatible plastic might not be visibly damaged but could incur changes in its chemical structure that embrittled or otherwise reduced its lifespan. I was working with a solid cast iron pan, so unless you know exactly what plastic the waffle iron parts are made of, the point to be careful with plastic handles or cords is well-taken.

That said, when using lye, think it would be far preferable to remove the iron component from the waffle iron and then follow the same procedure as many cast iron pan users have done, for the same successful result.

PS see the !castiron@lemmy.world community!