Payment processor Stripe, used by countless merchants around the internet, has confirmed that it will be integrating Apple Pay support into its system later this year. Web stores will be able to easily add Apple Pay as a payment option with their existing Stripe integration.
From Stripe:
The same fluid experience Apple Pay provides in iOS apps is now available anywhere on the web. Your customers can check out with just their fingerprint across mobile and desktop websites. Available this fall.
Additionally, Shopify has announced that all of its stores around the world will be able to add Apple Pay support at no extra cost. In fact, in order to integrate Apple Pay support, Shopify merchants will need only to head to their payment settings in Shopify, then select the "Activate Apple Pay" option when it arrives later this year.
Apple Pay on the web will allow you to pay for items in online stores both on iOS 10 and macOS Sierra when they launch publicly this fall.