Can a toilet paper holder be mounted to the side of an MDF bath vanity?
It shouldn't be a problem. MDF doesn't have the best screw-holding strength, but TP holders are very light so it shouldn't be an issue. If you are concerned, you could put longer (1") bolts - not screws - through the MDF and put nuts on the back, so you're not relying on the pull-out strength of the MDF to hold the screws.