Learn Dutch: Zonder de bon krijg je geen vergoeding. - Without the receipt you get no reimbursement.

Because the sentence starts with the prepositional phrase zonder de bon (without the receipt). In Dutch, whenever you put a phrase other than the subject at the beginning of the sentence, the verb and subject swap places so the main verb stays in the second position. This is the exact same verb-second (V2) inversion rule you use with connectors like desondanks.

Why does this sentence use geen instead of niet?

Geen translates directly to "no" when it comes before a noun, just like "no reimbursement" in English. You use geen to negate indefinite nouns (nouns without "the" or "this"). You only use niet when negating a verb, an adjective, or a very specific noun.

Can I use het bonnetje instead of de bon?

Yes! In everyday Dutch, it is incredibly common to use the diminutive form het bonnetje when talking about a store receipt. Both de bon and het bonnetje are correct and sound completely natural here.

How would the sentence look if I wanted to start it with je?

If you start with the subject je, you don't need inversion. The standard word order applies, making it: Je krijgt zonder de bon geen vergoeding. (You could also say: Je krijgt geen vergoeding zonder de bon.)