There should be an option if you click on either the pet or their portrait to Send Home (it's been a while since I've had this issue, so I don't remember)
That said, you should also be able to order pet supplies from the computer/mailbox (if vacationing in a residential world) or from either the kiosk (Granite Falls) or one of the marketplace vendors (Selvadordada). The only place I don't know how to get pet stuff is Batuu (and it would probably require editing one of the three lots.)