A website programmer is not required by law to have any qualifications, but some companies do ask applicants to have some sort of qualification. Many programmers are self-taught or learn on the job as this is often the best way to pick up programming techniques and skills.
You can chose to do a computer programming course at college or university which will teach you a good overview of many different programming languages. If you continue on to become a web developer then you may find that you only ever program in one particular language anyway so this is not always necessary.
Some web developers know little in depth programming and get by, by using code already created or by using open source software such as WordPress with can be edited quite easily. If you want a website created that is quite bespoke and requires complex functionality then you will need to find a programmer who is very experienced in that particular programming language and can write code from scratch.