How common is use of word "PhDs" to describe PhD holders in formal documents? The context is that this usage can be useful to describe people with PhD degrees while saving on extra characters in a formal document.

Example: "The position is typically reserved for PhDs" (instead of saying "PhD holders").

The question is relevant here to know how common is it for professors/academics to use this term in formal contexts without being frowned upon.
Extremely common, in both general usage and in my exposure to professors/academics. Unless you have some reason to believe there would be objections to it, I think you're anticipating a problem where none exists.

(On the other hand, "saving characters" is an awful reason for doing anything, especially these days when memory is cheap and printing is easy. If you use this form, use it simply because that's how people talk.)