Can I use the built in vacuum system for ventilation
Taking air out of the house like that would mean getting more fresh air into the house. A little would be good. A lot could result in drafts in strange places as the air would come in from wherever it can. The effect would likely be uneven as some areas, perhaps with leaky doors/windows, would get a lot of fresh air and others would get almost none.

In addition, the air coming in would be as hot/cold and humid as the outside. That would help feel less musty but not necessarily less damp or feel "good".

A quick search finds that central vacuum typically uses 2" ductwork where HVAC typically 6" or larger. That being said, perhaps an inventive HVAC tech. could come up with a solution to use the central vacuum ductwork for an HVAC system. Or possibly a mixture - use traditional HVAC ductwork for the easiest runs (and/or the rooms that need it the most, like the kitchen) and use the central vacuum ductwork for the rest.