If you have an HVAC system in your home, you know it's supposed to produce hot or cold air. But it's not supposed to produce water, and if you've got water dripping out of one or more vents, you know you've got a problem. Water coming out of a vent is not a good thing, but it is a common thing with three simple causes and fixes.
Part of the problem is due to the contact of hot/humid and cold. If you have cold air traveling through a duct, the duct material is going to become cold itself. If the air surrounding the duct is hot and humid, the moisture in the air will condense into liquid. This condensation is what you're seeing drip out of the vent. Normally, ducts are insulated well enough so that the cold duct material doesn't touch the warm air. But if the insulation is cracked, torn, or otherwise insufficient, condensation can quickly form. It is possible for the condensation to form on thin insulation, so then you have the added problem of soaked and mildewing insulation along with the condensation.
The fix for this is to simply replace the insulation. Ensure you have vapor barriers wrapped around the ducts as well as adequate new insulation surrounding all of the ducts. Have your whole home inspected instead of just the duct leading to the leaking vent because you never know if there are other insulation weak spots that are about to start dripping. You may as well check the whole thing and replace those sections once you find them.
A large part of that hot/humid versus cold problem has to do with humidity levels that are too high. One reason for this is that there isn't enough ventilation in your home, especially in the attic or crawlspace that hosts the ductwork. Clean out attic vents, have new ones added, and see if there are hotspots, such as from nearby water heaters, sending hot air up into the attic. You may also want to consider adding additional attic or crawlspace vents if your home needs them.
Even if your home has enough ventilation, you can still have problems with humidity just because of the weather. In this case, adding dehumidifiers may be the best option. There's not much you can do about the weather, but you can combat it somewhat.
If you're still having trouble with dripping vents or want someone to find the trouble spots that are contributing to the leak, call an HVAC repair company, like Atlantic Heating and Cooling. Try to find one that also does the insulation installation around the ducts. This problem with condensation is annoying and not one you want to let go, but don't worry. The situation can be fixed.