Doesn't make much sense to me that you have an admin capable of creating some additional "space" on your Linux, but not being able to change permissions to move some directories there as well.
That's simply the easiest way forward: If you want to move that directory, change the permissions to be able to do so and afterwards change the permissions back or alike. This is admin 1:1 your sysadmin should be capable of.
> The reason
> for this new storage area is so that it continuously gets backed up
> and is off the main area.
Keep in mind that Bugzilla might store data worth of being backed up in it's own "data"-directory as well. Things heavily depend on how you configured it, especially regarding where to store attachments.
> Additionally, it would allow others in our
> department who do not have access to linux systems to be able to
> access the bugzilla site,
Bugzilla is a web app, the only thing users need is a browser, so Linux is NOT a requirement for them. Moving directories around for MySQL doesn't change anything about who is able to access Bugzilla how. In theory it doesn't even change anything about how Bugzilla is hosted.
You are mixing VERY different things up here.
> again due to permissions and servers from the system administrat
> or. If anyone would be able to assist with this, that would be
> great. I am pretty new at this type of thing.
Your question is MySQL-specific, not Bugzilla, so it might make more sense to use one of their mailing lists. In the end, everythign depends on what exactly the storage is you intend to use for MySQL, why you have permission problems etc.
While you don't necessarily need to move the existing MySQL-directories, but e.g. instead copy them and change the config, you e.g. simply need to have permissions to change the config or replace the current MySQL-dirs with some symlink or alike.
Without permissions your options are very limited and OTOH there's a reason why you lack permissions. This means someone doesn't want you to change things as you would like, so at least you need to ask whoever denies you access why your access is denied and change that.