How can website access certificate store on my client machine?

Do web browsers allows websites to access certificate stores?

No, except in support for client side certificates (and the browser might have an option to always require some user interaction to confirm this), or—for Internet Explorer—ActiveX controls have access to everything the user has access to, this includes their certificate store.

In IE: Internet Options | Security | <Select Level> | Custom Level... | Don't prompt for client certificate selection when no [...]

I.e. IE will always prompt if there is not exactly one matching certificate, and optionally always.