|Poison Ivy is a very tenacious plant but can be managed with persistence.
It would depend on where the plants are located and how large an area you need to work on. If the vines are growing up trees or structures, simply cut a chunk out of the vine about 2 feet off the ground. Make sure that no part of the vine is left in that area you cut. I generally cut about a 4 or 5 inch chunk out and pry it out in these cases. The rest of the plant will die. You do need to monitor the plant though as the roots may try to send up new shoots.
Growing on fences? Simiar to above - you want to cut the vines near the ground. Growing on the ground? This is harder. You don't want to use motorized tools like weedwhackers or mowers as this will spew little bits of the plant into the air. In this case, I normally recommend RoundUp or a similar tough herbicide. As with all chemicals must be used in a prudent fashion (follow directions, cleanup well, do not overuse).
Now having a pet outdoors makes this more problematic if you choose to use chemicals. If the patch of poison ivy is small, you could consider pulling it by hand -- properly suited up of course! You have to decide how best to handle this situation. Could your outside dog be confined to an area where he couldn't get into any applied chemicals?
Check our "Control" section to see if maybe some homemade treatments may help eradicate the plant.
Hope this helps.