I am going to assume that each location already has web access. If you do not have access to the internet at each site, you need to get that sorted first. Then you will be ready for the next step.
Because once you have web access the easiest answer would be to skip trying to set up the physical hardware and then rolling your own software. The best way do the whole thing is to do it on the web.
Here is a rough outline of how I would see the project laying out
1. Get a domain name and set up an account with a web hosting outfit. (GoDaddy, Yahoo, Google, or a host of others).
Do your research. The company you go with should provide you with easily customizable "tools" (programs) that have all your basic features set up. You then customize the software to fit your needs.
You will want a store front, a reservation system, a secure credit card payment system, and a database to keep it organized. Look at the different offerings carefully. They will all promise to be "the best". Your job will be to find the package that is best for you.
You will not find what you are looking for at the entry level accounts. You will be paying a fee for these services. But given the hours and hours you can spend reinventing the wheel, and then debugging it, using proven software as a base can end up being a much less expensive alternative.
2. You set up the features you want on the web hosting site, separating public areas that anyone can view and secured areas were you can take reservations and accept credit cards. Another reason to go with a commercial web hosting outfit is that they have people that can help you do this. Again, it costs, but it's cheaper in the long run.
3. The biggest reason to use a commercial web hosting outfit is that they keep security people on staff, and handle the majority of security issues for you as part of the service. If you were to set up your own server, and wanted it to be accessed by potential customers (not just your campsites) you would have to set up your own web server and open it up to the Internet.
An improperly secured web server has a life expectancy of about 3.5 minutes before it is discovered. hacked and subverted. Having customer credit card data stolen is not a way to build clientele. Better to have some pros on your side.
4. With your domain up and running, you can access the system from anywhere, and do not have to worry about maintaining the hardware.
Paid WIFI is another story all together. Remember that typical residential hardware for WIFI only has a range of 300 feet and is not designed to be used outdoors. You can purchase commercial hardware, but bring your checkbook. This is where outfits like Cisco and Netgear make their money.
Once you figure out how you are going to cover the area so all the sites have a strong signal, you'll need to set up a computer with two Ethernet cards to act as the gateway for controlling access to the Internet.
There are plenty of "Internet Cafe" programs out there both free and retail for you to choose from. In general all work the same way. They monitor requests for web access coming from the campsite, and those computers that are authorized or have the password are passed through to the internet while the others are blocked.
The big deal is setting up the WIFI network. The rest is pretty cut and dried.
Hope that helps
Good Luck