Can I use 22/4 wire instead of 20/2 wire?
For power delivery through copper wires for this type of load the amperage capability of the wire is proportional to the cross sectional area of the conductor. You can check this in a standard AWG wire table. 

Your existing 20 awg wire has an area of .518 mm2. On the other hand a 24 awg wire has an area of .205 mm2. So two 24 awg wires @ .410 mm2 would not be equivalent to the 20 awg wire.

It is possible that your system would work OK with the pairs of 24 awg wires if the current draw in the wires did not come close to the total capacity of the originally specified 20 awg wire but I would suggest that instead you try to either find some actual 20 awg wire, use surplus wire that had paired 22 awg conductors or find 6 conductor cable with the 24 awg conductors and set them up in two sets of triplets.



Update. I misread the original posting as specifying 24 awg wire instead of 22 awg. Sorry about that. The fact still remains that two 24 awg wires in parallel are not up to the same current capacity as the 20 awg wire. However a pair of 22 awg wires, each with a with a cross sectional area of 0.326 mm2, will be more than the equivalent of a 20 awg wire. So now you know that a pair of 22's or a triplet of 24's will suffice as a substitute.