Yes. Ivy can be grown from stem cuttings so theoretically pieces kept moist might sprout and grow, but I don't know that I've seen this outside of a propagation setup where they were being looked after. The amount of trouble you might have with this would depend on what species of plant was being chipped, as well as how finely the chipper used cut the pieces.

Depending on the kind of chipper you use, the vines may wrap themselves around the chipper shaft.

I don't think they will chip well - although there will be shredding to some degree. If you have chipper, try it. If not, I wouldn't pay the rental to find out for sure if it worked.

You may get a lot of propagation from the small stems - not good if you don't want them.

Boy can I relate. I am working on a one acre site that is totally overrun with ivy and the homeowner hates it, so we are reclaiming it one section at a time. I would be very hesitant to chip ivy, can't imagine it would work well, would resprout, and ivy has chemicals in it that inhibit plant growth. Haul it to the dump. One thing that has been extremely helpful for me is hedge shears--cut it, slice it off and roll it up! Good luck and get a good massuese (sp?).

The following is a favorite tool around here for removing ivy and blackberries.

Here is a link that might be useful: woodmans pal

If you're renting the wood chipper and you run poison oak through it you're a criminal because it will give poison oak to the next person who uses it. The oils stay on the surface and will spead to the new chips. It takes barely a trace to spread

I'd worry about every little piece of ivy or blackberry rooting and exacerbating the problem. Since they are both noxious weeks, we just put them in bags and put them in the garbage, not the compost, nor spread on the ground.