Learn Italian: La fattoria aveva ogni animale che cercavamo. - The farm had every animal that we were looking for.

It is acting as the direct object. The subject of that part of the sentence is "we," which is built into the verb cercavamo. Since we are doing the looking, the animal (che) is the object receiving the action.

Why are the verbs aveva and cercavamo in the imperfect tense?

Both describe ongoing states or continuous actions in the past. aveva (had) describes a past state of being (what the farm possessed), and cercavamo (we were looking for) describes a continuous action without a clear start and end. The imperfect tense is exactly what you need to set this kind of background scene.

Why does the sentence use the singular noun animale after ogni?

The word ogni means "every" or "each" and must always be followed by a singular noun. This works just like English, where we say "every animal" rather than "every animals". To use the plural, you would have to say tutti gli animali (all the animals).

Where is the word for "for" in the Italian sentence?

You don't need one! The Italian verb cercare inherently means "to look for" or "to search for". Adding a word like per after it is a common mistake for English speakers, but cercavamo completely covers "we were looking for" all on its own.