I am developing a website that sells gift cards and at this point in time I am planning on selling only Amazon gift cards (in the beginning at least). I'm at the point where I need to pick a domain name and register a LLC. Would I be infringing any copyrights or trademarks if I named my company and domain name something like "Buy Amazon Gift Cards, LLC" and "buyamazongiftcards.com"?
A reasonable person might well believe that your enterprise is being sponsored by or is affiliated with Amazon, and so you would be infringing Amazon's trademark. Even if there is no reasonable confusion, Amazon might well think otherwise and take legal action against you. You don't want to start a business by defending a lawsuit, even if you win, particularly not a suit from your supplier.
Why not choose a different name, particularly as that would make it easier to expand into non-amazon cards later? Something like 'Buy Better gift cards' or 'Gift Card World' might be a better choice.
In general, you may not use a firm or product name that would cause reasonable consumers to confuse you with the trademark holder, or to think that you are endorsed by, affiliated with, or sponsored by the trademark holder without permission (which you are not likely to get).