Is there any plan from Stellar to allow website owner to accept payments with Stellar on their site?
There is no "official" payment solution, however there is a bunch of community projects.
(listed in order of public announcement on GalactickTalk)
A plugin for OpenCart to allow people to create online stores that can be paid with XLM or most any other of the many stellar.org available assets. The plugin also provides the option to setup escrow with multi sig that is built into stellar.
NoughtPay aims to help Cafes, restaurants, florest's and bakeries (many other) to advertise, calculate and process Stellar Payments.
PayWithStellar is a simple button that easily enables your website to receive payments in XLM Lumens, the currency used in the Stellar Network as the leading platform for world commerce.
StellarCheckout integrates e-commerce web sites with the Stellar.org blockchain allowing merchants to accept payment in lumens.