Learn Russian: Проект не даёт возможности, которой мы так ждали. - The project does not give the opportunity we were waiting for so much.

This is the "genitive of negation" which you are studying right now! When a transitive verb like даёт (gives) is negated with не, its direct object often takes the Genitive case instead of the Accusative. This rule is applied very consistently with abstract nouns like возможность (opportunity).

Is которой in the Genitive case just to match возможности?

No, it's actually a coincidence! The relative pronoun который must match the noun it refers back to (возможность) in gender (feminine) and number (singular). However, its case is decided by its job in its own part of the sentence. Here, it is the object of ждали (we were waiting for). Because we are waiting for an abstract concept, the verb ждать requires the Genitive case, resulting in которой.

Why does возможность end in -и in the Genitive case? Isn't that a plural ending?

It looks exactly like a plural, but it is definitely singular here. Возможность is a feminine noun ending in a soft sign (-ь). For this specific group of nouns (the 3rd declension), the singular Genitive ending is -и.

Why doesn't ждали have a preposition to mean "waiting for"?