There are many degrees and training courses that you can take to become a website programmer. You do not need any formal qualifications to do so, but many employers may favour those that do. If you are looking to become a self employed website programmer then these qualifications may be less important. Many customers look for experience and examples of sites you have created along with testimonials when choosing a programmer to develop their website.
Many of the best website developers are actually self taught. If you are learning in this way you need to be disciplined and ensure that you keep up to date with the latest industry standards and news to stay on top of your game.
If you are considering becoming a programmer then why not have a go in your spare time to start with as it does not suit everyone and you need to have a good knowledge base prior to starting up your own business.