Kids marking on wood? Raising kids can be tough!
Permanent marker can often be removed with isopropyl alcohol. First, assuming you still have the markers, try using isopropyl alcohol to remove it from a test surface. If it works, then try on the underside of your table to ensure it does not damage the finish. If it is safe, then go for it.
I just tried Goof Off on the underside of my table and it worked very well. Of course, the underside of my table is sealed with polyurethane.
You might just want to leave your permanent marker stain alone. This might seem like a strange attitude, but I resigned myself to marker stains years ago. Now, when I look around at the child-inflicted damage around the home, I think of it as the history of my family in this house.
You can also try some toothpaste, that is known to remove permanent markers on wood.
If your furniture is unsealed, the ink has likely permeated into the grain. In the end, it might come down to getting out the power sander.P