I don’t see why not.  Charities must use their funds for charitable purposes but sometimes those purposes are carried out by non-charitable entities (such as the for-profit utility that uses funds to provide free service to those who can’t afford it) or by individuals (such as missionaries in foreign countries).  A donor may not get a deduction by giving directly to those non-charitable entities that carry out the work, but will get a deduction for giving to a charity that retains the discretion to determine how, and by whom, the work will be done.  As long as you are not funding so many of your members so that the organization looks more like a mutual insurance company than a charity, and so long as the grants are for the benefit of qualifying animals, I think you should be fine.