I'm working with a new server and have run into a little snag.

I need to create a directory here: /var/www/vhosts/mydomain.com/

The server won't allow me to do this be default (permission denied error). How do I work around this?

Shell access.
Ask your host to create it.
Ask your host to set the permissions for you to create it.

Are some that spring to mind.

OP are you on the shared account or dedicated server and do you have root?

Do you have root access?

The workaround also depends exactly what you're looking to acheive... you might be better changing the DocumentRoot to be some subdirectory of httpdocs for example.

If your host won't handle this for you (or you don't have root access yourself) then another option might just be to use .htaccess for url rewriting. This is obviously not the preferred option as it can get complicated, degrade performance, and potentially expose more of a security risk than the other options. However, the exact impact and magnitude of these factors depends the specific characteristics of your situtation.

In other words: lots more info. please