Barbara:  Thank you for contacting the UVM Extension Master Gardener Helpline asking for advice on controlling Buckthorn and an unidentified vine.

I am attaching a link to the Vermont Invasives website, specifically the Gallery of Invasive Pants section.  This document shows photos of various invasive plants (such as Buckthorn).  Clicking on the photo will allow you to access more information about the plant.  The information includes identification tips, as well as best management options.

This document may also help you to identify your mystery vine through the photos.   (For instance, you may want to click on the photo of Knotweed or Mile-a-Minute Vine, to see if these match the description of the vine you are seeing.)

Gallery of Terrestrial Plants | Vermont Invasives (vtinvasives.org)

You also asked for a listing of contractors to help remove your invasives.  Below is a listing of certified invasive plant control contractors.  Contractors must be certified by the State in order to apply herbicides.  A certified contractor will also assure that the proper herbicide (chemical or organic) specifically formulated for the plant, is used. 

https://www.uvm.edu/sites/default/files/Extension-Community-Horticulture/Invasive_Plant_Control_Contractors_3-10-20201.pdf

I hope this information is helpful to you.  Good luck with your project!

Barbara:  Your photo was very helpful in identifying this vine as a wild grapevine.   Below you will find a fact sheet from Penn State containing more information on control methods.  They suggest leaving some of the vines in place because the vines provide shelter for wildlife.  Remove vines growing around valuable trees and leave some plants growing around lesser desirable trees.   One non-chemical method they recommend for shady wooded areas is to cut some of the vines at about a foot above ground.

This method may work well for you, since you mentioned that the vines are in a wooded area.

Wild Grape (psu.edu)