Yes, foxes can cause damage by digging shallow holes in lawns, as they forage for food.
They are attracted by the presence of worms, beetle larvae, caterpillars etc. If you have placed a fish, blood, or bone meal based fertiliser on your lawn, foxes will be attracted to this since it smells like buried food.
There are some things you can do to deter foxes from your garden. Do not feed foxes, intentionally or unintentionally. Keep food waste in secure bins, and provide secure accommodation for pets or livestock. Fill in any excavations with loose soil as soon as they appear to prevent foxes moving in. Prevent foxes from gaining access to your garden by making sure fences do not have any gaps.
Further information on deterring foxes can be found at the Fox Project