synonym for a "request to join"

Given a group of people and a person, if the group asks the person to join, the group "invites" the person, or sends "invitation." If the person requests to join the group, what is a word (preferably single-word) that means this, either in verb or noun form?

I have considered the following:

• self-invite: accurate, but is a two-word phrase and too colloquial
• impose/imposition: single-word, somewhat accurate, carries negative connotation
• appeal: single-word without connotation, but not very accurate

Solution 1:

apply

e.g. apply for a job.

"to make an appeal or request especially in the form of a written application" (Merriam-Webster)

(ETA: formatting and example)