This used to be in closed beta, but starting today if you have a meaningful YouTube following, you can signup to become an Amazon Influencer.
As an Amazon Influencer, you get your own vanity page on Amazon.com through which you can earn commissions on sales.
If you check out What's Up Mom's page on YouTube , you can see that she links to her Amazon Influencer page at the bottom:
Her Amazon Influencer page looks like this:
And she earns commisions on any sales she generates from that page. Amazon says they plan to open this program up to other social platforms including Instagram and Facebook soon.
Here's the signup page: https://affiliate-program.amazon.com/influencers
And more details from TechCrunch: https://techcrunch.com/2017/08/24/amazons-social-media-influencers-program-opens-up-to-youtube-stars/