Thank you for the quick reply! I really appreciate your help. :)
That looks to be a lot of work, doing a step-upgrade. They don't have a lot of posts on their blog, mostly pages. So maybe it would be easier just to delete the current version and make a fresh installation with the latest WordPress version and manually save and re-publish the posts and pages?
I am just worried their current template needs a lot of tweaks to work.
Edit: I exported their posts, pages and comments from their current version to another wordpress blog running the latest version. It worked just fine. So I think I will just delete the old version and install the latest WordPress.
Thanks for all the help!