Should I remove vinyl flooring to allow the floor underneath to dry out?
If you didn't put insulation and/or a poly vapor barrier over the bare dirt before you poured the basement slab, you made a mistake, and that mistake is causing moisture in the soil to wick into the concrete, keeping it saturated with water. That's very difficult to fix now, but vinyl will not mold, so you may be okay. If you're really worried about it, you might consider removing the vinyl flooring, sealing the concrete with a waterproofing agent like RedGard, and then installing new water-impervious flooring over it, like vinyl or tile--you can tile right over RedGard. Don't go with anything wood.

But there's more going on here than just that. The fact that your basement had a dirt floor and the house has no gutters leads me to believe that your basement was never designed to be finished; it was instead designed to harmlessly flood. Now that you've finished it, you need to fix the water management detailing of your house so that it doesn't flood anymore, because floods are no longer harmless. First, you need to add gutters. Then, you should build underground drainage channels for water so that it seeps into the ground a number of feet away from your house rather than right against the wall. The easiest way to do this is to insulate the walls of the basement as if you were building a shallow-frost protected foundation, which should also make the basement a bit more comfortable too. Trench down 1-2 feet all around the house, put rigid foam boards up against the exposed wall, and then place horizontal insulation sheets at the base of your trench, sloped away from the house. The result should look kind of like this:



Finally, drywall and wood moulding are bad choices for a basement that floods. Those are very moisture-sentisive materials. You need to implement my suggestions ASAP if you want to save them.