Why no Rudolph??

Remember that the movie is based on a book. We don't see Rudolph there, either.

North Pole Reindeer have been unionized since the early 30s. One of the negotiated points in the contract was limiting flight hours, including one year off for every 4. The crew that Santa had for this movie was the backup crew: Smasher, Crasher, Printer, Vickson, Vomit, Stupid, Stutter, and Blitzkreig. Rudolph is on the A squad so wasn't scheduled for that flight.

The "original" reindeers of the Santa mythos were the eight that were named in the 1823 poem "'Twas the Night Before Christmas" (Dasher, Dancer, Prancer, Vixen, Comet, Cupid, Donner and Blitzen)

Rudolph came to the lore much later, with a 1939 booklet published by Montgomery Ward, and gained wide popularity with the 1949 Christmas song we all know and love. The Rankin/Bass animated special came even later, in 1964.
(sources: https://en.wikipedia.org/wiki/Santa_Claus%27s_reindeer#Origins_and_history,
https://en.wikipedia.org/wiki/Rudolph_the_Red-Nosed_Reindeer)

Furthermore, (and this is my personal explanation) according to Rudolf's story he is not a regular, but an "emergency" reindeer, used only on foggy nights. In this movie, the night sky appears to be clear, so Rudolph was probably not needed and the elves left him in his quarters.

Somehow though I watched the Rudolph special as a kid, I believed in Santa but not in Rudolph. I was exposed to the poem The Night Before Christmas as a very young kid so I considered Rudolph someone's sequel to the original lore. Doesn't help the Rudolph special has Rudolph as Donner's son. And that all the other reindeer that actually make fun of him are the sons of Santa's original reindeer. It all just felt like a sequel to the original lore to me. But I know that's not every kid's experience.