I'm here with bad news for you, so I hope you can take some constructive and highly-realistic advice. ;)
First off, I applaud what you're trying to do.
Excel spreadsheets are great for simple tracking, for playing with the numbers, and as a platform for communicating data to others. On the other hand, using spreadsheets as a form of long- or even short-term data storage is a terrible idea for a myriad of technical and social reasons that your employers do not care about.
You need to face one truth immediately: there is nothing you can do to make them care about these issues. If easy access, better security/stability, or flexibility were important to them they wouldn't even have their current methodology.
So you need to change your approach, and the first step to doing it is abandoning your hopes for a moment where their faces light up and they just "get it". If you're telling them that the change is necessary and they're trusting you on that fact (even if grudgingly) then that's all you need.
It's no different than trusting your mechanic when he says you need a new part installed in your car. Are you happy about the expense? Of course not. The work you're doing and the subsequent retraining is an expense.
To address the other half of your question: you want them to be more involved, more proactive, in providing you with their "wants" regarding a new system. You want them to "think about what they need and how they work" and provide you with ideas about features.
I have bad news again. They won't and it's mostly because they can't.
They. Don't. Know. What. They. Need.
The truth is, the hardest part of most systems is the design, not the coding. And the hardest part of most designs is the interface. Design is a hard job requiring special knowledge and skill sets.
Do not treat design as the byproduct of coding. It's the other way around. Learn that well and learn it now. Almost anyone can learn to hack some code together. I've always felt my business card should say software DESIGNER.
Think of it this way. You hire a home decorator to come in and spruce up your living area. So the decorator comes in and starts grilling you for ideas on what you want the place to look like. Would that work well?
If you knew how to make your place look nice, if you had the ideas, why would you need the home decorator?
A good decorator would observe your living arrangement and how your carry about your daily business at home. He or she would make note of the way you've currently decorated, thus learning your general taste.
Then without your direct input, the decorator would put together an overall design. And only then would the decorator ask for customer input.
You need to do the exact same thing. For example, you know they like the spreadsheets. Well, what do we know about spreadsheets? They're dead-simple for data entry. They're very responsive (no waiting involved). They provide a very simple mental model through which to understand the data. They require no logins or authentication.
In other words, a spreadsheet gets out of the way and lets the person work.
Your new system, I would assume, is going to have a web interface that's far less direct, will require authentication of some sort ("I have to remember a password now?!"), will lag in response time, etc.
In other words, your new system will remove everything they like about the current system.
So you have to win them over in other ways. So figure out what frustrates them about excel, and how you can avoid that in your system. Figure out what data they have to manually enter into this log that you can automatically derive and enter in the new system.
In other words, you're going to make some things harder on them with the new system, that's unavoidable and you shouldn't be in denial about it. Your system will be better organized and cleaner; well what's easier, keeping your room tidy or letting it go to mess?
So with that reality in mind, find every single part of their tasks that you can optimize out so that your new system is a net gain. If you do that, they'll lose the grousing within a week or two and come to like the new system.
That's my advice, as someone who's built a career on doing exactly what you're doing: coming into a small business and replacing real-world system and outdated software systems with my own software.