The "fix" for this is probably going to cause problems elsewhere. XenForo's <xen:require> tags are not as robust as they could be (1.x did not support inlining, and did not preserve external order), which is why we used a different solution all this time. XenForo 2 improves it a bit, but still leaves much to be desired; while XenForo 2 now supports inlining, order of inlines vs externals is still not preserved.
I have implemented the change from VaultWiki's CSS externals to XenForo's built-in CSS externals on your site, which fixes your custom What's New page slightly, but I expect CSS on other pages to be broken now, due to the shortcomings in XenForo's externals system. Custom wiki icons will still not work on the What's New page, because XenForo 1.x provides no mechanism to inline the icon CSS over JSON. There is no way to overcome this limitation short of the What's New app not using JSON to load the results. So only the default CSS for icons can be loaded.
With some add-ons, you may also notice a similar effect with wiki-related Javascript not working in JSON output.