You guys are thinking constant Wattage.
For achieving the same "Wattage", the current at 24v will be half that at 12v. (Watts = volts x amps). You can't make the motor draw less wattage at higher voltage!!
That DC motor is a big honking resistance. As voltage doubles across a resistance, the current doubles.
I = E/R, and R is constant (for all practical calculations) so as E doubles to 24v, I (amps) does too.
If you want to try that winch on 24v, you can put resistors in series with the relay coils to limit the current to the coils. That will make your relay pack survive. The motor will still be a 'maybe' if you load it down.
As for doubling the load capability of the winch, it's mechanical system would not handle twice the load. (If the mechanicals could handle twice the load, the marketing geniuses would have called it a bigger winch. They already stretch the truth.)
Also consider that the Wattage dissipated in the motor, at 24v instead of 12v, will quadruple!! That sucker is going to heat up fast.
Watts = volts x amps, and both volts and amps are doubling (with resistance constant).
Now, in reality, the commutator in the motor cannot transfer all that power as well at much faster speed. The brushes will arc more and heat more, and that will limit current somewhat....for a short time.
Another option is to buy a 24v motor for your winch. You might need one anyway if you hook the 12 v winch to 24 v
Bob