The seats will dry out on their own, assuming you can leave the windows open at least part of the time. The carpet will dry, too, but more slowly. The padding underneath the carpet, however, is another matter altogether.
The padding is a sponge-like material that can soak up, and hold, water, almost indefinitely. There is no power on Earth, that I know of (no dryer, no Shop Vac, etc.) capable of extracting water from the padding underneath you car's carpeting. You have to do it yourself, or pay MANY dollars for someone else to do it for you.
Lift the carpet (not rocket science). Remove the padding. Let it dry somewhere. Reinstall the padding and the carpet.
Be happy that you saved lots of money.