If you use a primer called Kilz, it will cover it all up and then you can paint. Another suggestion re other things in the home such as the HVAC would be to canvass the neighbourhood and find out what differences are between your place and stick-built (plus neighbours might know more about what's gone on in your house than the RE agent is telling you).

I would wash the walls down well with TSP before using the primer. Also rinse. Will help to kill some of the smell too. Been there done that and I know it really helps. Wear rubber gloves.

Wash well first. You want paint holding only to smoke deposits? Primers can be tinted to the final color which might help in hiding and reduce the number of required coats of the finish paint.

FYI, for painting a tricky surface like vinyl wallcovering (or laminate cabinets & furniture) USE THE PROPER PRIMER!! Most paints/primers aren't formulated for sticking to vinyl, which is why some run into peeling problems. Your best bet is to use a primer that's designed to be a bonding primer, like Zinsser BIN or Zinsser Cover Stain. Both say they'll stick to all surfaces without sanding (though light sanding to scuff up hard-to-bond surfaces is always a good idea). Check the labels - most primers can't make that claim. Glidden Gripper primer or Sherwin Williams Adhesion primer are other good options. Kilz is a fine everyday primer, but it's not designed to be a super-adhesive bonding primer for tricky surfaces. Glad some of you have had luck using other primers, but for anyone just beginning this project, be careful and do a little research before you choose your supplies. Since you have a double-whammy of vinyl-hard-to-paint walls and smoke damage, I'd recommend Zinsser Cover Stain or another primer that can claim to stick to any surface while also sealing odors/stains.