I don't think anything will stick vinyl to something else and allow handling. I suggest that you add the feed chutes last, using something like Future floor wax or a high quality white craft glue like Sobo, to stick it in place. You'll have to hold it there until the liquid dries, but at least it will dry clear and keep things stuck together if there' s no load on the joint. Depending on the contact area, there's a rubber cement trick that involves letting a coating dry on the parts to be joined; they'll stick when you put them together - unfortunately, rubber cement will also dissolve paint at a touch.