For work stuff,docusign already solves this. Why are schools not on docusign? Budgets? For the same reason they may not subscribe to your potential startup?
I think the real problem is incentive: the school has no incentive to digitize the form - it is easy for them to just copy the old form and send it out each year. If we build a database of common forms and give parents/schools easy ways to digitize old forms, it gets them over the hurdle.
Plus, docusign, etc. don't build a structured database with the data to make it easy to access.
You're also talking about collecting personal data (contact info, medical info, etc.) digitally which brings on a whole mess of laws for this free service to deal with.