This is the original sentence:
While it had started out looking like a real disaster, thanks to Missy, they'd managed to avoid that.
When reading this I find it ambiguous (did Missy cause the disaster or help to avoid it?). (the context makes it clear Missy's action did avoid it)
What would you recommend?
1) a semicolon after disaster
2) an em-dash after disaster
3) remove the comma after Missy
4) something other (what?)
HM.
did Missy cause the disaster
Yes. Because of the two delimiting commas, the phrase 'thanks to Missy' can be removed from the main sentence ie 'While it had started out looking like a real disaster, they'd managed to avoid that.'
If, as you say, Missy's actions averted the disaster, the second comma is wrong.
AJ
The other option, is just to juggle the words around, a bit.
What was,
While it had started out looking like a real disaster, thanks to Missy, they'd managed to avoid that.
Becomes
While it had started out looking like a real disaster, they'd managed to avoid that thanks to Missy.
While it had started out looking like a real disaster, thanks to Missy, they'd managed to avoid that.
3) remove the comma after Missy
I prefer #3.
I'd change the "they'd" to simply "they."
#1 is wrong.
I would go with #4.
As a minimum, I would delete 'While' and replace the first comma with a semicolon. Then the sentence would read:
It had started out looking like a real disaster; thanks to Missy, they'd managed to avoid that.
The basic problem is one of awkward construction.
If spoken, the phrase is run on, but the normal pause in speech would remove the confusion:
"It had started out looking like a real disaster," he said, "thanks to Missy, they'd managed to avoid that."
But when written as in the example, the run-on sentence creates the confusion. The best cure is not the punctuation, but to change the bad sentence construction, i.e.;
"It had started out like a real disaster. Thanks to Missy, they'd managed to avoid that."
I don't have the answer other than to suggest avoiding the em-dash with anything online as special characters get messed up a lot...
requires immediate medical treatment.
Except for when the semicolon joins two related independent clauses.
I'm also going to suggest you might consider reworking the beginning of the sentence. Aside from the punctuation issues which others have addressed, the main verb in the first clause is looking, so that should be what Missy has avoided.
In other words, the sentence could be read as Missy changing the perception of the accident, not its severity.
Another way of writing this without the verb confusion is:
"It could have been a real disaster, but thanks to Missy they'd avoided that."
Same general structure and vocabulary, but removing the source of confusion.