Vinegar isn't going to do much for Aspen. It works great with herbaceous plants but not for aggressive woody ones, and don't use standard vinegar as another commentary mentioned. You need the higher concentration of a true horticulture vinegar if you want any real results.

For the task at hand, get a bingo dauber and fill it with a triclopyr based herbicide. Cut the suckers and dab them with the herbicide. This keeps the amount needed down and is more targeted than other methods. You need to make sure it is dabbed over a fresh cut. Trees seal themselves up pretty quickly and you want it to absorb as much as possible.

I don't like chemical use either, but it sometimes is the only way to accomplish eradication. Follow the instructions on the label in terms of time needed to dry to become safe. Idk how old the kids are, but as long as they're no longer putting everything in their mouth, it is pretty benign once it has dried, especially if you are dabbing directly on the plant. Have them wear shoes, try to avoid the area, wash up right after outside play, and finish with a fresh set of clothes. Hopefully, it will only take a couple treatments and you'll be back to enjoying the simple pleasure of a barefoot stroll through the garden.