Certified GKH Gardening Expert
The label on a name brand lawn insect killer says it also can be used on the vegetable garden of listed vegetables. It tells how many days to wait till harvest.
Usually the vegetable garden is a good distance from the lawn and may even be in a raised bed. Vegetable roots don't go that deep and if the lawn is a safe distance, I wouldn't think vegetable roots would intermingle with grass roots. If you have grass roots in the veggie garden you may want to be sure all your veggies are listed on the label as safe to use around.
Remember, if you use bug killer on your lawn it will also kill earthworms, which are beneficial.