Have you thought about using one of the standard web tools like Google Apps, and just putting your data in a shared Google Sheet? Easily shared and no real costs involved.
Good call, that's exactly what I'd recommend, and if you wanted a little more polish, you could use Google Forms to easily build a nice UI, and provide basic validation for data entry. Then have that data dumped into a Google sheet that you could distribute, do some analytics, etc.
I say this as someone with years of software development experience.
On the other hand, if you want to explore the other option, you know for fun, then there are plenty of service options under S3 that will give you the tools you need.
[doublepost=1491307738][/doublepost]In the simplest model you'll need a service for the app, and a service for the DB, these can be on the same server, you can just spin up a small EC2 instance on S3, then start adding packages to support your design target. Probably some combination of opensource tools on a Linux instance, the Amazon AMI is nice, easy starting point. Then decide on a web framework, Rails, Nodejs + Express, Python + whatever, even (ugh ...) PHP. That will give you the server side processing and database access core, then add a DB instance (personally I'd use Postgress but from your perspective vs. mySQL, either is more than fine).
When you say this:
I'm pretty good with programming websites
I'm assuming you mean the browser side, i.e., markup, CSS, maybe Javascript (vs. any server side experience)