You could use either glyphosate or triclopyr beneath the trees to control the poison ivy. 

For both it is important to follow the labeled recommendations. The benefit of triclopyr is it is often more effective for poison ivy control and when used as directed it will not kill the grass. However, triclopyr can be absorbed through the bark of the trees and cause injury, so care would need to be taken around the trunks and exposed roots. There is a bit more info from an older MSUE article here https://www.canr.msu.edu/news/be_careful_where_you_spray_that_stuff

Here are some tips for glyphosate use that may be helpful...with the most pertinent items underlined and bold. 

When using products containing glyphosate there are a few important points to consider. First, as with any pesticide, remember to read and follow all labeled instructions. Second, glyphosate is a broad-spectrum herbicide, meaning it will injure or kill other plants contacted during application, so care is needed to avoid green plant material, exposed roots, and injured bark of desired plants. Third, glyphosate is relatively safe in the environment when used as labeled. It adsorbs strongly to soil in most cases (i.e. clay and organic matter), allowing even sensitive crops to be planted shortly after application; meaning no carry over issues are expected. Fourth, glyphosate alone can take up to 14 days to show full activity under ideal growing conditions. Retreatment of the area may be needed depending on the degree of infestation. Glyphosate is most effective for perennial control in the fall but can be applied anytime the plants are actively growing (temperatures consistently above 50F). Finally, be sure that the product you choose has only the active ingredient glyphosate or glyphosate + pelargonic acid. Products with additional active ingredients may have other unwanted effects and may delay the planting of other plants in the coming season(s).

One other thing that I have in my mind is that you'll want to see what the residue looks like after poison ivy control. The irritant urushiol can persist on dead tissue for several years, so you'd want to make sure it is out of there before people sit beneath the trees. 

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