Baking soda also works- but not that the same time as vinegar.
Lysol laundry disinfectant works well also.
Won’t help with a laundromat, but while looking at new washing machines several had settings for running without clothes to self-clean the device.
FWIW, there are cleaning pods available for washing machines (“Affresh” is one brand). They’re not normally necessary, and, again, not useful if you’re stuck with a laundromat. But it’s likely that the machine itself is the problem.
On a perhaps unrelated note, my normal detergent is (or I should say, was) Tide Original. But I need high-efficiency (HE) detergent and when recently I couldn’t find Tide Original in the HE formulation in the size I was looking for at my local store, I bought Tide Ultra Oxy instead. That stuff is great and I’m using it from now on. It gets out stains better than the regular stuff. Might be worth a shot.
Definitely concur with letting the condo association know about the problem, as it does sound like the machine itself might need cleaning. And vinegar can work wonders in many household cleaning chores. It’s absolutely amazing at dissolving hard-water scaling, for example.
Thinking about laundromats, the only stuff I take to a laundromat are things I’d rather not put in our home machine. Horse blankets, dog vomit encrusted blankets, towels used to clean up nasty stuff.
I mainly use vinegar if an item in the load was there because I dealt with a cat accident. It takes urine smell right out.
Can you try washing a load in a different machine–maybe another laundromat in town? If the funk remains, you’ve eliminated one possible cause.
Been meaning to ask, so I’ll just piggyback on this thread if ok…
If I wash my sheets but let them sit folded in the closet, after several days they smell to high unholy hell, welp enough at least to make it tough to sleep in them. #1, why does this happen, and #2 how do I get rid of it/stop it from happening in the first place? My current solution is to air them out by blowing a high-powered fan on them after they’re on the bed, works pretty well after 2 hours or so.
Is your closet smelly in some way? Like moth balls or something?
Is it dry and mildew free?
Cotton or synthetic fabric?
If it’s the sheets I’m willing to say they’re not getting clean in your wash cycle.
Bleach baby bleach.
And any smell good stuff you can live with in the dryer.
Good question. The OP said perfume is a no-go but does that mean no smell good things can be used that smell good?
I have never tried but I’d think there are other alternatives to perfumes to make the clothes smell better (a satchel of herbs maybe?..guessing…NO idea if that would work).
I want NO smells at all-I am very sensitive to such and they’re guaranteed to keep me up.
Bleach I’ll look into.
Yep. I almost feel bad sometimes. But not really. 
I have no experience owning laundromats but I think unless it’s oily or you’re dye-ing things they probably consider the machines self cleaning.
I do have experience with lots of laundry washing and home machines. They are not self cleaning.
Coin-op it’s catch as catch can. Look inside. If any visible gunk is present try another machine
Look on the bulletin board for complaints.
Eavesdrop on patrons.
At home, follow your manufacturers instructions for cleaning the machine.
Personally I’d wash my dainties in a bucket before I’d put them in a coin-op machine. (Yep, that’s Victoria’s secret… shhhh!)
Modern washing machines may have a self-cleaning cycle you can run. Works pretty well but it is not 100%. You still need to work a bit at it for best results but it’s not too bad (mostly seems getting rid of hair which is always gross no matter whose hair it is).
I run a self-clean maybe once every 1-2 months and do a “deep clean” once every six months…give or take. Works for me, YMMV.
Piggybacking off the OP: My laundry has a similar issue, but it’s inconsistent. Half the time, it comes out the dryer with a burnt-rubber odor. The other half, it comes out fine. I thought at first something must be damaged in the dryer machine that’s causing the heated-rubber odor, but then why would the odor be absent in the other batches? Each batch is of similar weight, detergent, fabric type, etc. and the heating cycle is always the same. The lint trap is always cleared.
Run a few tests. Wash a normal load then air dry a few pieces and put the rest in the dryer. See if there is a difference (may need to run the test more than once to be sure).
If it is a coin-op place make sure you use the same machine. Some may be good while others might be bad.
I’ve been using borax in the wash for years. It helps soften our local hard water and acts as a laundry booster.
I have also heard to use vinegar but have not tried this myself. nor have I tried spray bottle of vodka, although I did hear a podcast that recommended it. not the one at the link, but I googled “laundry guy” and he’s the one. I think it was on public radio’s lifekit.
same guy says people are using way too much detergent, like with those pods.
I read that residual detergent after the load is finished contributes to microbial growth and funky smell. Running an extra rinse cycle, or a whole extra cycle without any detergents or other stuff added, may fix it.
You mention no bleach. Are you sure all bleaches would be a problem? Or is it just chlorine based bleaches that are a problem? There are oxygen based bleaches that don’t have any chlorine – might they work?
My laundry machine choice is restricted, because I do not care to spend $4-5 per load at other "mats. (This is a college town, & they screw the kids out of every nickel.)
I will give a cup of cleaning strength vinegar a try, this Washday Sunday.