Hire some pros if you know you or your family members are highly allergic to the poison ivy oil.
If you attempt to do this, watch YouTube videos on how to.
I would dress up as I was going into a BSL4 lab to work with Ebola. Lol! Just kidding, but not too far off. Wear double layer clothing with long sleeves (prepare to toss the outer layer in the trash), tuck your long pants into tall socks, shoes should be washable (rubber boots would be best), wear long, sturdy rubber gloves (toss them afterwards), good face mask, hat, face shield or goggles. Just make sure no skin touches the plant and you are not breathing in any dust or oil from cut plants. Do not use weedwackers or anything that could create a cloud of poison around you. Cut the plants and dispose in sturdy trash bags (not compost!!), then either dig up the roots (tedious!) or paint the stubs with brush and weed killer. The plant easily re-grows from roots, so those need to be killed - mulch over the area will not help. Watch for any new growth and remove as soon as you see it. The soil will likely be contaminated with the oils, so I would let the area sit under mulch for a while before letting anyone play in that area. Make sure to clean your boots and all tools with rubbing alcohol and throw the cleaning rags in the trash. Operate with the presumption that anything that touched the plants is highly contaminated! Good luck!
For digging up roots, the best tool is a shovel appropriately called the Root Slayer.