This is a hard question and software people will typically not want to be pinned-down on specifics without consulting with you extensively, because the depth of requirements for software projects vary dramatically. You're asking for very little functionality (which is great!) but the bulk of costs of software construction are not in the core functionality, they are in polish, bug-fixing and edge-case handling, which you included in your requirements ("simple, secure, and easy to use") but would need to be a lot more specific about, in order to get a result you'd be happy with. It's really hard to give these kinds of estimates without those conversations.
To just satisfy the requirements, I could imagine doing the non-payment-processing version very cheaply using Typeform or another "survey builder" for the UI, and a simple backend that just pulls the results out of Typeform and sends notifications. I could imagine throwing a prototype for this together in a few days of time for someone like me (so less than a few thousand $) but it would probably not work well, be reliable, etc. at that level of polish. Also, few freelance developers of quality will take on a project like this at that price point, because they have "seen it all before" and know that the edge cases and polish are likely going to result in a 10x cost inflation at a minimum.
Payment processing increases the cost a bunch because it adds new requirements for security, ease of understanding & debuggability, etc. You can often outsource the actual UI for it to e.g. Stripe Checkout though so it may not make a huge impact.
I don't think you've given enough information to answer "what would be the best way to go about it?" For example it sounds like you need software help, but you might also need graphic design help, product design/UX help, etc. For software help I know that freelance developers will bid on your projects on various sites (I'm not familiar with which sites are currently considered reputable and I haven't done this myself so I don't know how hard it is to figure out which bidders are any good). But it's unlikely you'll produce a "simple, secure and easy to use" product without getting help in other domains as well. It is sometimes possible to convince people to work with you for free if they like your idea (and you).
Thanks so much for the quick and detailed reply! I agree I haven't provided enough info yet to let you (or anyone) answer the question well. This is mostly because I came up with my real idea very recently, and I'm a total rookie in this field so almost everything is an "unknown unknown" to me at this point. Also, for now I want to avoid giving too many details, until I've worked out the idea more fully.
But to address some of your points and provide some additional info that may be helpful:
What they said. Also, for the specific example given, consider:
You may want to read the tweets by Patrick McKenzie about his call for something similar and how it worked out: VaccinateCA.
Thanks for the recommendation - very glad he is doing this. My favorite part was this paraphrased dialog with a pharmacy:
Us: How would someone make an appointment with you?
Pharmacy: Go to the county website at...
Us: You know you're not on that, right?
Pharmacy: WHAT.
Inadequate equilibria at their finest. I wish him all the best in his efforts to make them more adequate.
I'm not a programmer, so I have very little idea of how much it costs to, for example, write the code for a website or app. Suppose I wanted to hire someone to design a very simple, secure, and easy to use app - how much should I expect to pay in terms of time and money? To work with a concrete and timely example, suppose I wanted an app that would do the following:
Note that I'm not suggesting that such an app would make much of a dent in the logistical nightmare that is our healthcare and political systems. I'm using it as a hopefully easy to understand example of the level of complexity I'm thinking about.
As a follow up, how much additional cost would be needed to process payments through an app? (Again, not thinking of this in terms of the vaccine example app, but asking about apps in general).
And finally, if I did want to eventually hire someone to write an app, what would be the best way to go about it? Would people in this community be willing to help?