Translation:We leave the animals with the organization.
I suppose when you translate it as 'at the organization ' you're dropping off the animals for good....like an animal rescue organization. If you use 'with the organization' you're dropping off the animals temporarily ... like at the vet or dog show. Yet really, 'at or with ' would work. The problem is with the word 'organization' It's too ambiguous as are many of Duo's sentences. For Duo's ambiguous sentences, a native speaker would be asking himself 'What country is this guy from? What an odd sentence '