From Curt Danhauser's Guide to Animated Star Trek:
The reason Chekov wasn't on the Animated series was money.
[...] At first,
Filmation planned on not having George Takei and Nichelle Nichols come
back to do their roles again. But when Leonard Nimoy learned of their
exclusion he said that he would "... not be a party to this if two of
the minorities who contributed to making STAR TREK what it was when we
were on television cannot be incorporated."
It was due to Nimoy's
stand that Sulu and Uhura's characters made it into the animated
series. Unfortunately with that many star voices, the budget simply
didn't allow for Walter Koenig to return as Chekov.
There were some conflicts on the series. Nichelle Nichols’ and George
Takei’s cartoon likenesses were being used but they weren’t originally
hired to provide the character’s voices (Paramount owns their younger
faces, apparently). Hoping to save money, Roddenberry decided to have
Majel Barrett voice Uhura and James Doohan do his best Sulu imitation.
Only a promised work stoppage by Leonard Nimoy got them on board.
"It is common knowledge that Ensign Chekov was cut from the Trek cartoon series for budget reasons. To compensate Walter Koenig, he was allowed to write one of the episodes. However, Chekov was seen in the cartoon pilot episode "Beyond the Farthest Star" in about three scenes which take place on the bridge - you can clearly see for a few seconds Chekov sitting at the helm with Sulu, only to have it cut away and come back with the three legged alien, Arex (voiced by Jimmy Doohan) sitting in his place. I used to have 16mm film print of this episode so I was able to examine it closely, but it is clearly visible in the episode."