vBulletin Manual - Using Forum, Blog or CMS in a Subdirectory.

Here are the steps to move your forums, blogs or CMS into a subdirectory other than your root.
Forums and Blogs -

Copy directory and rename do_not_upload/blog and/or do_no_upload/forum to the directory you want blog or forum to be. For example do_not_upload/blog to /my/vb/dir/blogs

If this is not a direct subdirectory of the vbulletin directory, edit vb_dir.php and change the $vb_dir path to point to your vbdir (this is necesary if you want the blog at myvbulletin.com/this/is/my/blog/blog.php for some reason)

Edit the "Forum Component Url" under "Site Name / URL / Contact Details" and/or "Blog Url" under "vBulletin Blog: General Options" options (new options) in the settings to point to your new directory. These can either be absolute urls or relative to the vb root url. Note that at present urls on a different subdomain are not supported (they will play havoc with Ajax and possibly introduce other issues).

If you are using mod_rewrite friendly urls, you'll need to change the .htaccess code to handle the new directory. A sample for where the blogs are in blogs and the forums are in forum:

RewriteRule ^forum/threads/.* forum/showthread.php [QSA]
RewriteRule ^forum/forums/.* forum/forumdisplay.php [QSA]
RewriteRule ^forum/members/.* forum/member.php [QSA]
RewriteRule ^blogs/blogs/.* blog.php [QSA]
ReWriteRule ^blogs/entries/.* blogs/entry.php [QSA]

At present, the "Always use Forum as Base path" must be set to "yes" when using the subdirectory feature. You can find this option at:
Settings -> Options -> Site Name / URL / Contact Details

Note, that CMS can already be moved into a subdirectory. This features has been available for several releases. Instructions can be found under the CMS setting help text.

Subdomain Support
Due to the way that AJAX communications in vBulletin is implemented, and how browsers restrict those communications in browsers, this is not possible at this time. That is to say that you cannot access the forums from https://forums.vbulletin.com/ and use AJAX features at the same time.