In STII: TWOK, we see Chekov as First Officer of the Reliant but that's as far as his on-screen career went. Shouldn't he have gotten his own ship by the time of ST:VI or even Generations?
On the other hand, Sulu still seemed to be a helmsman in ST:V and then one movie later, he's Captain of the Excelsior. We never see him serve as First Officer, which would seem a logical step even if it wasn't required.
I wonder if the events of TWOK put a damper on Chekov's career path? Maybe he was indirectly blamed for some of the things that happened and kept getting put at the end of the promotion queue.
You're overlooking the obvious. Nothing in TWOK would have affected Chekov's career. Khan's escape was unfortunate but could have happened to anyone who visited Ceti Alpha V. The likely reason for Chekov's inability to gain a command of his own is due to his participation in Kirk's hijack of the Enterprise
in TSFS. The formal charges against the former Enterprise
crew were dropped in TVH (except Kirk's insubordination charge), but that doesn't preclude Starfleet holding off on any promotions for the foreseeable future. Whatever Kirk and his friends accomplished, the UFP wasn't likely to let them get off almost completely unscathed.
Sulu finally received his captaincy in 2290, but there's no telling what he had to do make that happen. Original drafts of TWOK had him already being promoted to captain of the Excelsior
: the Enterprise
training cruise was supposed to be one last joy-ride before he took command of the vessel. This was dropped by the time the film was produced, but the novelization has a scene depicting Kirk congratulating Sulu on his upcoming assignment. That his ascension to the center seat was delayed suggests Starfleet wasn't entirely forgiving of everyone who aided Kirk in his efforts to save Spock.