How could I simplify or shorten this compound sentence involving "for which"?

It's fine, though a bit stilted (and "awhile" is an adverb: you mean "a while").

More normal would be "another institute I ordered a transcript for at the same time". But I don't think you can do it "beautifully" in a single sentence because of the the amount of information you want to get into that clause.

The sentence technically is grammatically correct, although you're right, it does sound a bit confusing. This is likely due to the amount of information you're trying to condense into that single sentence.

You could split up the sentence, and say "I had ordered the transcript a while ago. Another institute already received it last Thursday".

Or, you could keep it in the same sentence but reword it as "The transcript was ordered a while ago, and another institute already received it last Thursday".

In a general case, don't re-reference a subject you've already discussed earlier in the sentence. In this example, you already said "The transcript was ordered a while ago", and if you're continuing in the same sentence it is implicit that when you are referencing another institute it is on the matter of the transcript.