Learn Polish: Proszę o odpis, aby udowodnić pochodzenie. - I ask for a copy in order to prove my origin.

Aby is a more formal, written-register equivalent of żeby. You will often see it in official documents, academic writing, or formal requests like this one. In everyday speech, a Polish person would almost certainly say żeby udowodnić. Both mean "in order to" and are followed by the infinitive here because the person asking and the person proving are the same.

Could I use kopia instead of odpis here?

You could, but they have different legal weights. A kopia is just a standard photocopy or reproduction. An odpis (from odpisać, to copy out or transcribe) usually refers to a certified copy or official transcript, like a birth or marriage certificate, which is exactly what you would need to prove your origin (udowodnić pochodzenie) to a government office.

What case is odpis in this sentence?

It is in the accusative case, because the phrase prosić o (to ask for) always requires the accusative. Since odpis is a non-living masculine noun, its accusative form is identical to its nominative (dictionary) form.

Do I always need a comma before aby?