Looking for the title and author of a sci-fi short story wherein an Earth spaceship with telepathic crew encounters an alien spaceship. The telepathic crew try to convince the aliens that the ship's cat is in charge...
This is almost definitely "Trouble with Treaties" by Katherine MacLean & Tom Condit as mentioned in this previous answer. It was originally published in 1959 in the anthology Star Science Fiction Stories No. 5 so it definitely qualifies as "vintage."
It's been a long time since I read it, but as best I recall the (human) telepath figures out that the aliens think the humans aren't forceful/dominating enough to be as apparently advanced as they are, and unless there's some evidence of a more powerful race in charge of the humans, that the aliens will simply take them over. The humans hatch a plot to pretend that the ship's cat is secretly in charge of them - and the cat so powerful that the humans don't even realize it's in charge.