I am using the S508 Refresh with WCAG 2.0 in it. Do I need to support the current browser or can I design to browser in particular, like Chrome? I have a personal view about it and I just want to make sure it is correct and where it says that all browsers need to be supported. I need the support documentation for it.
Yes. It's the intent of Guideline 4.1:
The purpose of this guideline is to support compatibility with current and future user agents, especially assistive technologies (AT). This is done both by 1) ensuring that authors do not do things that would break AT (e.g., poorly formed markup) or circumvent AT (e.g., by using unconventional markup or code) and 2) exposing information in the content in standard ways that assistive technologies can recognize and interact with. Since technologies change quickly, and AT developers have much trouble keeping up with rapidly changing technologies, it is important that content follow conventions and be compatible with APIs so that AT can more easily work with new technologies as they evolve.
In other words, you need to make sure your website can be interpreted by a wide range of user agents, including widely used web browsers.
The only legitimate reason (or perhaps "excuse") for supporting only one browser, is if you have full control of the context where that user agent appears.
An example might be some kind of information kiosk or console in a museum, library or other public place.
If you content is to be generally accessible over the web, or even over an intranet by multiple devices, this will not apply, because the web is intended to be accessed by a plurality of user agents, whose main common features are web standards such as the WCAG specs.
If you're making content for only one browser, it's not bona fide 'web design'.
If you exploit the new-fangled features of a particular browser, but don't prevent other user agents from getting the same essential content in other ways, that's completely ok, as long as the proprietary features don't break anything in the other user agents.