First congrats on having your rental paid off in a year. Unfortunately you are running into a problem a lot of landlords with no reserves run into when issues like this come up.
This kind of put you at the edge of becoming a slumlord. Personally as a landlord I am going to protect my investment. I would at least get the issue looked at by an independent party. Not the tenant because who knows what happens if you let/tell/suggest the tenant handle this. I don't want them taking off vent covers and guessing this and that.
This could be nothing, or could be something beginning. I want to know exactly what this is and get it taken care of. I would not want to live in a place where I smelled what I thought was an odor and nothing was being done to see what it is. No telling what is causing the odor until it is properly checked out.
When was the last time the HVAC system was checked out? Start there.
Thanks for your reply. The air conditioner was serviced a year ago and the furnace a couple weeks ago. Never had vents cleaned except the dryer vent once. I do supply furnace filters to encourage and require changing once a month. I wouldn't usually allow self repairs but I know the vent cover comes right off and don't see that much harm can be done to try a simple vac. But Murphy's law says the tenant will find a way to break it, right?