Just wondering if there's a feature on Wave where once I've paid my bill to a vendor that I can generate some sort of payment advice or receipt to give to the vendor to say that I've made the payment?
No, there isn't. It couldn't count as proof that the bill was paid since recording a payment to a bill in Wave doesn't actually move any money. It's on the vendor to provide a receipt for a payment.
Can I ask in what circumstance you would like to have a document like this? Maybe I can offer an alternative or a suggestion?