"I want proof."
All of my experience in Greece has been this word used to mean "receipt" (like a simple receipt as opposed to a "timologio" - the one that gets submitted for a tax refund.
Yes. Proof and receipt use the same word in Greek. That's because the receipt is a proof of the transaction. It's more common to use Θέλω αποδείξεις (in plural) when talking about proof, though.
I'm wondering why there is no article. My (incorrect) answer was θέλω η απόδειξη and instead it suggested the correct answer of Θέλω μία απόδειξη. I'm not understanding why it would be "a proof" instead of "the proof".
Your answer was incorrect because you used the definite article in the wrong way, using the nominative form "η" (while nominative is mainly used for subjects). You should have written "Θέλω την απόδειξη", since the word "απόδειξη" (proof) is used as an object. Besides, when there is no article in Greek, it means that you do not refer to a specific thing (like "απόδειξη" here), so this is why Duolingo suggested "Θέλω μία απόδειξη" (using the indefinite article) instead of "Θέλω την απόδειξη" (using the definite article). However, the translation "Θέλω αποδείξεις", when talking about proof, sounds more natural in Greek, as troll1995 pointed out.