I see a bunch of games providing users with the ability to earn Amazon gift cards and other real-world rewards. 

What is the best way to integrate this functionality? Are there good third-party providers that offer some sort of plugin that makes this a seamless experience for my users...

The first stop with that is your lawyer.  Any time where you give real money or things of tangible value to players, the developers must ensure they are on the correct side of gambling laws. 

Violating gambling laws -- even if you didn't realize you were violating them -- can land you in prison.  You seriously need to discuss with a lawyer before proceeding.

Thanks, but I'm just in idea mode right now. I'll reach out to a lawyer when I have done more research.

I found something like this: https://pointvoucher.com/About "Consumers can download exciting branded mobile games and play their way to real products and experiences for the time they spend interacting with the games. Vouchers can be claimed in the branded voucher shops connected to the game or in the Pointvoucher shop where they will be exposed to games from other partners."

The experience is a bit complicated in the app though. You have to separately create an account on pointvoucher, which I'm not a fan of.

Any other ideas people?