Here's a link to a picture of a wall tap.

https://www.amazon.com/s/ref=nb_sb_n...words=wall+tap

My question is why don't they make these that will
screw into a GFCI receptacle like the one in the link
instead of just plugging into the existing?

I've seen a version at Home Depot that even have the
reset button on the front of the tap in case it trips
but none that are 'permanently' mounted.