Can web design and accessibility live together? Can they coexist? Only under certain conditions. The first condition concerns the choice of the color palette. Color contrast, that is, the contrast between foreground and background color must be accessible. For accessible, I mean that it must follow and fulfill all the W3C requirements that are clearly expressed in the Contrast Analyser tool. This is a no-no condition. For example, the following image shows a wrong color contrast.
In this case, white and orange cannot be used together. Period. But if you don't want to throw away your own design, there's a simple solution: provide an alternate design to be delivered to users with special accessibility needs (such as blind-color users or users with some other color impairment or even low-vision users). This solution is up to you: if your site must be accessible for whatever reason (e.g. legal, like Section 508), then you have to consider this solution. Otherwise, it's only a matter of personal choice.