So you think that your legal vendor systems only need the internet to work if your use a SaaS (Software As A Service), like GMail? The answer is "no", many applications use the internet to function, even though they reside on your PC desktop.
Some vendors use the internet to check your licensing of the software, and if your system does not verify the "license request" the system will stop working (granted they have a grace period before shutdown). A lot of the collection systems from big consumer groups, have a live link to their main accounting systems so that you can work with live data when contacting clients. Instruction systems like home loans instructions, need the internet to send new matters, or process payments. And what about your communication via email, these days people expect an answer when they email you.
What I am getting at here is the fact that you need to have an internet connection link to the software provider.
This is a very important question to ask your vendor, whether they have a redundancy plan if one of the three/four major Internet Service Providers (ISP) go down, which will allow your office to continue working. MyBroadband has an article that touches on the condition of one of the big three's network called "Recent DC outage reveals potential network problems", a good read to understand that the servers hosting the legal applications, should be mirrored between 2 big ISPs - to minimise the risk for you the customer. Basically, if you cannot access the server via one ISP link, you will be able to access the server through a different ISP link - it will cost the vendor a bit of money - but in the long run I think it will benefit their business, as well as the law firms.