Yes, that primer has quite the odor!!

Actually, VOC's & ODORS have ZIP to do with the other! Some low/no-voc Waterborne paints smell kinda BAD!

There's nothing to do but wait. Provide as much "Air-exchanging" as you can tho'. Oil-base products don't "dry out" in the traditional sense. They CURE be reacting with Oxygen molecules in the air.

The more air you can "exchange" per day, the better. Put some fans in windows blowing OUT. Put cardboard or something around them. Have other fans pulling air IN.

Good Luck!

https://iaqa.org/about/

Hire an indoor air quality consultant.

Based on findings -

Find out if mitigation is covered by your home owners insurance

Discuss findings with seller - ask to have them assist with cost of mitigation.

Ok, thank you to everyone! I am doing the blowing air out of basement suggestions with cardboard helping to ensure air blows out.

There is no carpeting in the home. There is some kind of laminate used in the kitchen, and its new, I'll try to figure out what it is. I asked a neighbor and they did say there was mold in the house (they had been in there before it was sold to help the owner), but it isn't clear where. Though I'm unsure mold would cause VOCs. I suspect that's why the Kilz was used in the first place - to kill the mold? There is no obvious mold on the boards/vents in the basement - nor was it found in the inspection.

In terms of air quality mitigation, sure I'll try that, thank you. I've never heard of insurance covering that, but I'll ask my agent. I've talked to the seller about this already, and he said he'd bring in ozone to treat it, but I'd have to remove everything from the home. But does ozone actually stop VOCs? I thought that just stops odors, but that's not primarily what I'm worried about.

I agree w/ @live_wire_oak. My experience with Kilz is similar and no or only very limited odor after 2 or 3 days. I'd look for something else. In the meantime use exhaust fans in windows, ideally in the affected area or as close as possible in a way that pulls fresher air through the offending area.

There is a problem that has been going on for at least 10 years with some paints that don't cure properly and stink forever. We repainted our guest bathroom in 2013. The paint dried but stunk like chemicals. We waited a whole year and the smell did not go away. I found some very extensive forum links from people who had the same problem. The problem seemed to be across paint brands. Just did a search on it and found this forum with posts from 2019/2020 so it is still going on.

https://www.bobvila.com/posts/89017-lingering-paint-fumes

The fix that most resulted to on the forum I was following in 2013 was to replace the drywall. Which is what we eventually ended up doing in 2015. Tore the whole bathroom out (tile also due to there being drywall above the tile line) and replaced every painted surface with new drywall. Smell gone!

Many people tried encapsulating the smell with various products, baking it out, ozone machines etc. We tried baking and ozone with no luck. From the many posters it seemed only drywall tear out and replacement worked so that is what we did. I was very nervous to paint the new drywall and every paint brand you could think of was on the thread as having caused problems.

Zero VOC and low VOC paints seemed to be the common thread of people that had problems and it was across all brands. I suspected improper mixing of chemicals from China that were common to all paints. As a result I chose a paint made in the USA called ECOS paint. I spoke to them on the phone and they said they did all their own manufacturing and had not had the problem with the odor that doesn't go away.

https://www.ecospaints.net/

Edit: Just found this thread that sounds like your situation with Kilz.

https://www.reddit.com/r/legaladvice/comments/6rz6qq/apartment_has_constant_smell_for_11_months_mgmt/

did you find fix? same problem here. was painted w kilz primer underneath by previpus owner.

did you find a solution?