MSU Extension has a helpful publication on this topic: https://www.canr.msu.edu/pestid/resources/plant-and-weed-identification/PoisonIvy_E3438_AA%20(002).pdf

Poison ivy is a native plant that should be left to grow in natural areas, but should be controlled in areas where pets and people frequent to avoid painful rashes. Here are some key things to know for management:

Nonchemical control options are limited:

• Poison ivy is a perennial vine that had a taproot and rhizomes, removal by hand takes persistence
• When removing the plants, take care to cover exposed skin. Wear waterproof gloves. Bag plant material and disposeof it with the trash. 
• Never burn poison ivy as the allergen, urushiol, is volatile and can be inhaled. 
• Do not mow poison ivy as airborne particles may reach uncovered skin or be inhaled. 
• Wash all tools used in eradication efforts to remove any urushiol. Urushiol does not break down quickly and can remain on dead plant material, gloves, and tools for over a year. 
• If you have a lot of poison ivy or it is hard to reach, consider hiring a certified professional to remove it. Many lawn services, landscapers or arborists can help with removal of poison ivy. You'll just need to call and ask. Your local garden center is a great place to ask for professional references. You can also look for a certified arborist in your area using https://www.treesaregood.org/findanarborist/findanarborist .

Chemical control options:

• The two most effective herbicide active ingredients for poison ivy control are triclopyr and glyphosate, though 2,4-D has also been shown to be effective. 
• Triclopyr and glyphosate are most effective when carefully applied as a concentrate to cut stumps (if large enough, drilling holes in the cut stump also aids in uptake). 
• You can apply in any season, except early spring when sap flow may reduce movement to the below-ground tissues.
• Applying the herbicide as a concentrate to the cut stump(s) can minimize any off-target risks. 
• Glyphosate is a nonselective herbicide and can kill or injure any nearby desirable plants if contacted, whereas triclopyr and 2,4-D are selective and will not kill grasses. 
• Triclopyr is often more effective than glyphosate in controlling poison ivy; however, it can persist in the soil for up to 30 days or more depending on the environment and can therefore potentially injure sensitive species planted into the treated
  area within that time. 
• Glyphosate does not pose a risk to re-planting once dry due to binding with clay and organic matter in the soil. 
• Use caution with these products as they will kill or injure other plants with which they contact green foliage, green bark (non-mature), injured bark, or exposed roots. Triclopyr can also be absorbed
  through mature bark.

Always read and follow the label when using any herbicide. If you are attempting to control a large population of poison ivy, consider hiring a certified professional. Remember that dead poison ivy plant tissue still contains urushiol, and therefore, take protective measures if removing it.