I haven't bought any legos in a long time, so I guess I don't know exactly what parts you're talking about.
Usually when manufacturers say vinyl, they mean polyvinyl chloride (PVC), so I would think that you could get an appropriate glue at the hardware store.
If the plastic is high-density polyethylene, then I don't think there's any way to solvent bond it. I had to buy some acrylic from a distributor once; while we were there, I asked about some HDPE that he had on display, but he said we couldn't glue it together; it would have to be mechanically fastened with nuts and bolts. That, plus it was more expensive, made us stay with acrylic.
I doubt the pieces are Teflon because teflon is very expensive and overall has poor mechanical properties... it doesn't wear well.
If you can't get it to work using solvent or glue only, you might wan't to try drilling holes in the things you're trying to glue and then use some epoxy. The holes should be wider at the bottom than at the top, preventing pull-out of the epoxy.