Clare,
Sharepoint is essentially a collaborative tool - a tool for sharing various kinds of data. It is certainly possible to use it to host a shared membership database, but it's a complex product, and would be overkill for such a relatively simple application. It also requires a fair amount of technical know-how, plus you need either a hosting company that is able to support it or the technical skills to host it yourself.
It seems to me there are two possibe appoaches you could take:
1. Look for a product like Zoho - or one of a dozen similar products. (Try a web search for on-line membership system, perhaps.) These are entirely non-technical; they're easy to set up; in most cases, you don't need to install any software and never any hardware; and you can be up and running in a couple of hours. The main disadvantage is that you are limited in the amount of customisation that's possible.
2. Develop you own system based on a common database, of which MySQL is the best example. MySQL is readily available, free of charge, from most hosting companies. It is well supported, and will easily cope with your requirements.
The disadvantage of this approach (and it's a big one) is that you will need to develop some software to run your system. A database like MySQL is merely the storage. You also need a way to enter your data, query it, keep it up to date, generate reports, etc. In short, you need a front end. You don't need anything complicated - possibly just a few web forms if your needs are simple. But don't make the mistake of thinking that you can set up a database and immediately be able to access and update your data.
If you have the necessary technical skills, and are willing to invest some time in developing the front end (or some money to pay someone to do it),option 2 will give you a better product.
But if you just want to get up and running quickly, option 1 is a better bet. (Note that I'm not specifically recommending Zoho, but it is a good example of the class of product I have in mind.)
Hope this makes sense.
Mike