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Don't try to jump too far at once
The initial points that struck me when I read your question were that you are young, with no apparent qualifications, and with little experience that is relevant to the role you are trying to move into. Having said that, you have 18 months' helpdesk experience, so you clearly have something to offer, in both knowledge and determination.
These are the aspects that your CV and letter (or e-mail) of application need to emphasise, together with your keenness to further your career by part-time study, which again illustrates determination. If you feel that your CV style might be letting you down it might be worth your while investing in one of the excellent books on the market that shows you how to construct a targeted and effective CV.
However, you do seem to be trying to make too big a jump at once, and very few companies would be willing to take you on and train you virtually from scratch towards your new career. My advice would be to try to gain a helpdesk position with a larger company that has significant involvement in Web-based applications.
Look for a company that has a good career development policy (maybe one that runs the BCS Professional Development Scheme or Career Development Framework). Let your new company see your worth while you are studying. Keep up your efforts to expand your experience of the Web or programming, and hopefully the company's career development policy will allow your managers to see your potential and move you in the direction you seek.
Solution by Sheelagh Flowerday, British Computer Society
The panel: MSB International, Best International, BCS, Computer Futures, Computer People, Elan, Reed Computing, Netheads Consulting, Zarak Technology, E-Skills NTO