After periods of very heavy rain over several days, water comes up in the floor vents of my HVAC system in my sunken living room. We have to then syphon it out with a wet vac. We have a French drain system, but this does not always prevent the problem. What can we do to fix this problem in the least expensive fashion? Is there a permanent fix to the problem?

Sounds like you ducks are in the concrete floor and I have never liked this arrangement. The expensive fix is to fill the ducks with concrete and install in the wall or overhead ducks, way way too expensive and messy. I would suggest you consider digging around the perimeter of your foundation to about 2 feet. Lay down some gravel and then perforated plastic pipe with a cloth sock, then more gravel. I'd also install a sump pump pit 2 feet deeper than the tile and discharge at least 15 feet from the home. Lots of lablor but only about $300. Be sure to slope your plastic tile toward the pit with a drop of about 1" every 4 ft.