Learn French: Qui que ce soit, il demande une invitation. - Whoever it is, he asks for an invitation.

The phrase qui que (meaning "whoever") always triggers the subjunctive mood because it expresses uncertainty or concession. soit is the subjunctive form of être (to be) matching the subject ce. So, qui que ce soit literally translates to "whoever it may be".

Is qui que ce soit a fixed expression?

Yes, it is a very common set phrase meaning "whoever it is" or "anyone". While you can use qui que with other verbs in the subjunctive (like qui que tu sois for "whoever you are"), qui que ce soit is by far the most frequent way you will encounter this structure.

I want to say "asks for". Why is there no pour before une invitation?

The French verb demander already means "to ask for" when followed directly by a noun. Adding pour here is a classic English-speaker mistake; you just need the direct object, une invitation.

Could we use elle instead of il in the second half of the sentence?

Yes, if you already know that the unknown person is female, you could say elle demande. However, when the gender is completely unknown, French defaults to the masculine pronoun il to mean "he" or "the person".