Hello...poison ivy can be difficult to control but you do have options.  You can start by pulling the smaller, localized plants by hand...like from the sidewalk cracks. Try to do this after a rain and it should be easy to get the full vine out. Be sure to wear protective clothing and place the vines in a bag to dispose of in the trash or find a spot in the yard where they can be isolated to dry out. Be aware, though, you can still pick up the oils from the plants even after they are dead.

For the larger vines that are on the trees, it is best to sever the vines growing up the trunk to eliminate the flow of moisture from roots. Cut out a piece of the vine (maybe 6 inches or so), then use a brush to paint the cut ends with a shrub/vine
killer herbicide that is labeled for poison ivy. This way the herbicide is only absorbed by the ivy vine and not spread beyond that. 

If you decide you want to spray the ivy you can't pull by hand, you would use a herbicide which has glyphosate and/or triclopyr. It is non-selective so it will kill other plants that it touches, but, it does not persist in the soil.  You would want to do this on a very calm day so that you can spray only the ivy leaves and not have the herbicide accidently land on other plants.  See the website below for more information on the herbicide. 

If you use any herbicide, be sure to follow all instructions on the container. 

Below are links for more detailed information.  The Maryland Grows site actually talks about a situation very similar to yours. Hopefully this is helpful. Please write back if you have any questions. Robin