Why won't the "Employee of" field autocomplete when used as part of a edit profile form in Wordpress
It's on purpose, to protect privacy of your contacts. What this autocomplete does is to provide the list of all the organisations that you have in your CRM.

I could easily imagine quite a few organisations that are in the database, for instance of your "opponents" that would look weird, or even embarrassing in the list of suggested organisations.

And even it if weren't the case, it's not us, as the software provider, to decide what contacts in your database can or can't be displayed to anonymous visitors, so the only safe option is to remove that feature

I have written a while ago an extension to let you add an autocomplete, but pretty sure it doesn't work with the latest version of civi (that has switched to a more modern autocomplete jquery module)

If you have dev skills or budget, I could dig it out so it could be used as a basis for the feature (and so you could only provide your members in the autocomplete for instance)