I read it about ten years ago but I think it was written quite a while earlier.
He has a lot of trouble looking after the elephant in zero gravity.
He finally gets to the customer and tells him he has brought a specimen of Earth's largest mammal as requested.
The customer does not accept the elephant. He had been expecting a blue whale!
Very likely "An Elephant for the Prinkip" by L. J. Stecher, Jr., described in Goodreads as "A fun story about hauling an elephant from earth to a remote planet. As you can imagine there is a lot of detail to deal with. Told in a very fun and light hearted way." Published in GALAXY, August 1960, where I read it 64 years ago but don't remember any more about it than the Goodreads synopsis. But here it is: https://galacticjourney.org/stories/6008galaxyanelephantfortheprinkip.pdf .
