I'm not sure if I have the terminology correct but what I'm trying to figure out is if I can save my current VPS setup running on Ubuntu 14.04 so that I can transfer it to another hosting provider. I'm running Nginx and HHVM and I would really prefer not having to reinstall that and more (HHVM was a headache). Is this possible?
No, there is no universal way to do this. Each provider has their own way of taking snapshots, and very few of them give users access to either the raw block devices or the snapshots themselves.
This is reason number 173738 that you should be using a configuration management system to configure your servers, not doing it by hand. If you'd done so, moving to another provider is trivial.
I'd recommend starting with a CM system like Ansible. Start building play books against a local Linux VM, and iterate on those configs until you have things working as desired. At that point, deploy the new server and copy your data from the old system.