In casual writing and speech, who're works wherever you'd write who are: Who're you meeting later? Save the full form for formal writing, as explained below.
If you're drafting something formal, spell out who are so no reader mistakes the contraction for a typo.
Who're is a legitimate short form of who are, listed in learner dictionaries.
Situations and their sentences
Texting a friend about plans: Who're the people we're going to see? This is the Cambridge Learner's Dictionary example. It compresses who are into one word, matching what you'd say aloud, so the message sounds like your voice.
Messaging a coworker: Who're we waiting for? A good working habit, since the contraction mirrors spoken wording.
Casual group chat: Who're coming to dinner on Friday? The contraction mirrors how the words sound when spoken, so the message reads like talk. The form itself is standard English; what varies is only the tone of the message, not its correctness.
Emailing a client or writing a report: Who is attending the conference? Spell it out here. In formal or client-facing writing, the full form keeps the message professional, and it avoids the contraction being mistaken for a typo, since some readers may see who're as too chatty. When in doubt, the full form is the safer choice.
One qualifier that changes the decision
The one qualifier that changes the decision is formality. If your sentence is headed somewhere formal or evaluative, like a report, application, or anything being graded, spell out who are instead. In those settings the full form reads as careful, edited prose, while who're sounds like spoken or casual writing even though it is a real contraction.
Casual message to a friend: Who're you inviting to the party? Nothing here is being evaluated or published, so the contraction reads as your natural voice.
Neutral written context: Who are you inviting to the event? In a report, application, or anything formal, the full form is safer because some readers may see who're as too chatty or mistake it for a typo.
The usable test is simple. If you could say the sentence out loud to that same reader without adjusting your tone, who're fits. If you'd naturally switch to more careful wording, switch on paper too: Who are the organizers?
The dictionary evidence only goes so far. It confirms who're as short for who are but doesn't rate contexts, so the formal-versus-casual habit above is a writer-facing recommendation, not a rule the entry states.