Learn Dutch: Het jaar dat we de trouwdag hebben georganiseerd, is mij altijd bijgebleven. - The year that we organized the wedding day has always stayed with me.

Yes, you can. Het jaar waarin is grammatically correct and means the exact same thing. However, in modern Dutch, it is very common to replace prepositional relatives like waarin or waarop with a simple dat when referring to time words like jaar, dag, or keer.

Why is the verb is placed immediately after the comma?

This is the classic verb-second (V2) rule in action. The entire relative clause chunk Het jaar dat we de trouwdag hebben georganiseerd acts as the subject and takes up the first position in the main clause. The main verb is must come second, so it appears right after the comma.

I know blijven uses the auxiliary verb zijn, but does the separable verb bijblijven always use it too?

Yes. Just like the base verb blijven (to stay), the separable verb bijblijven (to stay with someone / to be remembered) takes zijn in the perfect tense. That is why the sentence uses is mij bijgebleven instead of heeft mij bijgebleven.

Why is it mij and not a prepositional phrase like bij mij?