Dynamic web forms: find a builder or make from scratch?

Making the forms is pretty easy. Passing data around is much less easy. I'm not sure what services are best for hosting dynamic forms yourself, but I think it would probably take you awhile to learn enough to do this from scratch.

Gravity Forms. Period.

A designer friend of mine who prefers not to code has had good luck with Formstack. However, if you don't have a deadline and are curious about programming, building a system that moves user input into and out of a database via a web front-end is the ideal first project. It is not trivial, but if you find the challenges rewarding to complete it could very well hook you on developing an increasingly valuable skillset. There are obviously many (many MANY) ways to get started, but I'll go ahead and get the ball rolling by suggesting you look at https://www.djangoproject.com/. It's heavily built on the concept of forms, abstracts away much of the nastiness of databases, and is written in Python—which is a great language, and an especially great first language. The Django tutorial assumes a knowledge of Python, HTML, and basic HTTP fundamentals; however I pushed through it before I knew Python and found it a decent introduction. Good luck!

For the first use case, many companies have taken a stab at making it easy to construct these lightweight business applications over the years, but nothing has really stuck in a big way, especially for on-premises deployment. I suspect there's a lot of these written in Visual Basic with Word or Excel. I did run into somebody who is still making a living in part doing Filemaker projects, which I had no idea still existed but actually might be quite a good solution for your problem. We often whip up something quick in PHP. The article you link to has quite a few of the online form builders i've seen/used. They're quite handy and 100% what you want, but make your IT Compliance people cringe because your data, often personal data, is living somewhere unknown and outside their control. Not sure I recommend for government work. For the second use case, check out note-taking software like Evernote or cousins like OneNote. People have developed all sorts of cool tagging strategies for these. To-do software is another popular category that could work for you. There are a thousand of them and they do all sorts of tagging magic. You might also like something like Trello, which is a lightweight kanban board/listmaking/tagging/todo/project management sort of thing.

The first thing to do (well, after you nail down your business requirements) is to figure out your technical infrastructure. Business requirements might include things like: security audits, whether the form needs to be accessible outside your company intranet, if it should be accessible in different languages, how many people will access it, and so on. Technical requirements might be "our company intranet is a custom .NET installation and this should run on top of it" or "our website is externally hosted on a *nix system, and the primarily customer-facing website is really a Wordpress blog with an existing Gravity Forms license" or "our web team has hacked together a custom PHP webapp for our intranet." Figure out if your form (really a super-simple webapp) can live on top of your existing technical infrastructure. (Hopefully so-- if you're at the point of asking "how do I write a dynamic form, you probably don't want to be the person to set up and maintain your own server.) Once you know what your requirements are, you have a much better idea of how to take your next step. But first I ask this question: do you want a form, or do you want to learn how to program? Because if the former, it's probably much faster to find your tech/web team, and ask them to do it for you. If the latter, there is much fun and probably frustration in your future. :-)

Seconding Gravity Forms. It's amazing and easy to use. It requires WordPress too, in case that's not clear.

The government website doesn't include any WordPress aspects to it, as far as I've seen. Most pages have static content, and I know some users are tied to an Adobe product to update web pages. I'll contact the IT folks to see if they have any in-house capabilities to add/build dynamic forms, but from what I've seen, that's not too likely. Our site (and the vision for our site) is pretty dated and not too inspiring, but I'll keep my fingers crossed. As for my database of notes, I'll search for to-do software and look into Evernote some more. But if I was to look into coding, where would I start? Django sounds interesting, but I only know basic HTML. Then again, I'm interested in learning Python for GIS applications, too, so I might head down that path. Thanks!