Grey -- I believe you just note in the etsy item listing (and if you want also your shop announcement) that x% of the money for the sale will go to X charity.
And then you would want to promote that/those items in teams, outside of etsy, and in your listing tags (eg. add a tag for the listing that reads "animal charity" or "animal donation" etc).
.....and then give the money to the charity. I don't know how it works re: taxes/ legality. I often say "100%" of sale donated, and then I pitch in the minus etsy fee and paypal fee...but I don't record it as a tax deduction. Sometimes I've done "50% of profit" etc.
Good luck!
Will help promo your listings if you create them!