Is there any way to put drawers on my big plastic toolbox? Answered

Nested stacking trays might be a better solution and probably easier to do. You can make them so they just lift out or you can get fancy and do them like a fishing tackle box where they are on rails and lift out when you open the box.

I found a picture of one here.

I don't see why you couldn't. Glues, screws and pop rivets will all work in the appropriate applications.