There are a number of ways to host a website.
As you say, you get a small amount of webspace with your ISP and this can be used but you will need to access it with your broadband password and user account details which you say you do not want to give out (sensible)
Another option is to pay a web hosting company, like 1and1.co.uk, fasthosts.co.uk, webfusion.co.uk etc to host the site for you. They charge for this but if you shop around you can get special offers.
If you know someone PC savvy they can set up your own webserver to host the website but you need a PC that can be left running 24/7, 365 and someone who knows how to maintain and secure it.
You will also need to buy a Domain Name and it should be agreed with the Club what that should be, something like yourtownridingclub .co .uk and this will cost as well but the same applies, shop around. You can get them for £5 for 2 years then you have to renew it. This will also give you personal email addresses as well like yourname @ yourridingclub .co .uk