How to invite multiple users when editing user account?
You question consists of actually 2 questions:


... have this field on my user/edit page.
... enter multiple e-mail addresses separated by , .


To answer the 2nd question (= ... enter multiple e-mail addresses separated by , ), have a look at the issue about Multiple emails support for Invite form. The patch contained in comment nr 5 of it can be used to 
create a new invite entity for each email address. Attention: it seems that this patch still needs to be rerolled for the latest dev version (which is also what explains the "needs work" status of this issue).