I agree with the majority on wording if you do this, but I also don't get the point of asking if gifts will be exchanged this year. Were you planning on asking each individual, or does someone make a master decision on it?
I'm asking because if you ask and the answer is yes, then aren't you obligated to spend money yourself on others' gifts? It seems to me it might be easier to just inform them you're opting out of gift giving this year.
I've done this twice and, oddly, both times it was for massive and unexpected vet bills. And our family is really close - nobody would think to ask if gifts were being exchanged, because the assumption was of course we're giving gifts!
So maybe that's an easier option if you want to keep money on tap for your furbabies.