You get call and request cards from villagers when you craft their ?special? furniture (like the electric bass for Apollo or the sports car for Cherry) - but since these are very rare events, the cards are extremely scarce, too.
Do you mean the animals at the campsite? Every hour, three of them offer a single request (at least from what I have observed). The roaming villagers offer three new requests everytime they move to a new location (every three hours).