I think for something like this to work, you'd have to get approval for a project and the scope/timeline before anyone is on-boarded. I'm fairly fortunate where I work, but a lot of times, getting buy-in can take long and be a rather painful process.
Also, do you have a scope of what type of things can be built? It would be beneficial to work out of a common Github/etc. so that future projects can build off of previous, and updates can be rolled out this way. For example, I know an organization I used to work with wanted a simple reception app - schedule an appointment, and it sends out reminders via text and/or email on the timing, and then it can also be used to check in. This would be simple using Twilio/Tropo and could be super easy to save in a state for other orgs to plug in and start using.
Great idea by the way - excited to see where this goes :)