How do teams of animators access the same master folder of Illustrator files?

Your request is a little bit confuse, probably because of the technicity of your request.

First some preliminaries:

• The issue here are not the Illustrator files but the After Effect files (the film data). They will tend to be quite huge.
• You need to determine if you are working locally or remotely (or a mix of it). This will largely influence you technology choice.

Some technical explanations:

• VPN is a virtual private network. This means that when I work remotely, that I connect via a standard internet line to my office network (or even home network...). As that standard line is an open line and anyone listening could intrude, you use for such operations a special software, creating a kind of data pipeline to a place. As long as you work in that environment, your data passes through that pipeline back and forth as if you would work inside of your office. For doing this, you need a fast internet, as you add to the load the VPN stuff. And when you do browsing on the internet, you do that via the corporate infrastructure ie sending a request via the VPN to the company and the router sends that out on the internet and returns the answer via the VPN line back to your PC.
  
  VPN is only needed, when working remotely or when connecting different offices around the world with a corporate network (I see servers in China, the USA, Italy and elsewhere in the world).
  
  VPN is neither cloud storage nor NAS. It is only a technology to make your local network securely accessible over the internet.
  
  
  
  
• (Inter)Network storage or commonly called cloud storage like Adobe cloud, Amazon cloud and similar, open you to store data on remote storage systems and to access them via the (inter)net. So this is a kind of outsourced storage solution, but also working over the internet, using a secured line, similar to a VPN or your home banking.
  
  
• Local network storage or NAS (network access storage) means that you buy a computer that serves as a server, one or more hard disks and put some sophisticated server software on this device. You can use either an out of the box solution (ideal for 2-5 people: Qnap, Synology,...) or you use full fledged servers (like Linux servers, Windows servers)  needing more maintenance but giving also more possibilities. Just to say: Qnap and Synology are also running Linux, but you will not need to know the OS to work with them.
  
  You will connect to those NAS drives like local drives, except that if your LAN is slow, you will have a penalty. But in general internet storage is slowest (therefore it is cached locally), next is NAS and then there is direct connected storage: your local hard disk or SSD.
  
  

If you are working locally you need NAS. If you are working remotely you need either NAS and VPN or cloud storage big enough to hold your data.

In all cases, you need a backup strategy.

Well just to say: you also now need a backup strategy, especially as an external hard drive is probably not the most reliable.

Did you try to share a Creative Cloud folder?

Does your company use networked drives or maybe Amazon Cloud / S3 Bucket storage?