Sounds like my back yard! The thicker stems with bark are wild grape. The leaves that are still green are catbriar, a thorny-stemmed vine with underground rhizomes. It's not poisonous, but you may also have poison ivy. All are chippable, but you'll need to treat the regrowth with herbicide (brush killer) a couple of times.

Steve,

Found a huge thread on wild grape in the woodland section. You were correct thank you. I also spend two weekends ago removing the posion ivy. Didn't feel like dressing up like a Ghostbuster again this past weekend so I went after the wild grape vine. The other thread some posts said not to rip it out but it has weaked a couple trees to the point where they need to be removed and is going into my Cedar trees. In my yard the Cedars rule. Sorry Wild grape. Since you are right on I have another vine bad there that has run right up the tree with smooth bark. Looks like it could be poision ivy but it is thick enough where it should have hair all over it and that is not present. I need to cut this tree down too but left it up while I figure out what type of vine is on it.

Poison ivy is smooth and can be huge. One of them was two inches thick and growing up, 40 feet, on a pine. Being very allergic, I sprayed as many flourishing leaves as I could reach with a herbicide, waited about 6 months after its death to remove it, hoping the rushinol poison had deteriorated. I left the ugly, dead vine, being cautious. Gravity has taken almost all of it down and I have thrown the pieces into the woods to rot completely.