Build some sites, gain experience
You have good, transferable skills. However, there are already a lot of people who want to get involved with Internet work and it is not easy to make a move into a new area. To get the job you need experience, to get experience you need the job.
A potential employer is more likely to be interested in you if they can see that you are planning your career move and you are taking positive steps to achieve it.
Start by gaining experience of Web technologies. Register for a course. This need not cost a fortune - a number of adult education centres offer courses in Web design. If you sign up for a course that leads to a qualification you are likely to be eligible to buy a student license (usually at a significantly lower price than the regular license) for some of the better Web design packages.
Design your own site. Volunteer to create a site for the village/swimming club/football team. Get experience. Compare your sites with those you find on the Internet. Consider becoming a reviewer for one of the Internet-based companies that specialise in providing feedback on Web sites.
When you send out your CV (try speculative letters) stress your transferable skills, say what you are doing to achieve your goal and give the URLs of the Web sites you have created. Initially you are more likely to be offered a business analyst post, but if the business is based around Web design you are on your way.