I've made an Amazon purchase for my business (self-employed) solely using an Amazon gift card which I received for non-business reasons. Does this still qualify as a business expense on my tax return?
Method of payment should have no bearing on whether something is a qualified business expense or not - it may make auditing more challenging, but it's not required that expenses come from a "business account" or from gift cards given explicitly to the business.
Now if the gift card came from someone you do business with, that may muddy things a bit - meaning you might have to record the gift card as income as well, or have documentation that explicitly states what the card was for (if it is not business income).
You could just as well bought something for yourself with the gift card and then used your "cash" to make the business expense, and you'd be in the same spot in the end. Or you could have "sold" the gift card to the business and used it. All that to say that the source of funds is not relevant to make something a qualified business expense.
Just make sure you keep the receipts.