Electrical – Why does the GFCI outlet trip when a vacuum is plugged in downstream

Check for a neutral to ground short somewhere on the load side of the GFCI and downstream outlets. Also, make sure there isn't a shared neutral with any other circuit and that you don't have a connection from the neutral load directly to the line side of the GFCI (in other words, the whites cannot all be pigtailed together, the line and load neutrals must be separated by the GFCI).

This sounds highly unusual some plug mounted gfci s need to be reset whenever they receive power but as you said you can unplug it and plug it in without having to reset it so I would check the wiring at the switch to make sure that it is breaking the hot wire and not the neutral also make sure the bare ground wire is not contacting the neutral at the outlet or in the switch box.