Was a NH resident for years (recent ones, even). Can't think of a single time I had to interact with a state agency that required proprietary software. It would stand out if I had. The closest I can even think of is if they're using Zoom for remote hearings? Never had to have a hearing about anything. This seems like a solution in search of a problem.
There are many web-based things that technically require Internet Explorer, even though other browsers seem to work fine. The real issue is likely that most of the things were built with IE in mind and only formally tested on IE. I feel like the real first step is simply to audit the functionality of these things in various browsers to see if they actually work properly or not.
Suffice it to say this problem doesn't require a government solution.