I would say no you don't need the full web address, but I would say 'the Prospects website'. The proposition would be 'on'. Also, I'd change the first part 'of the graduate trainee' to 'for a graduate trainee', thus:
"In response to your job posting for a graduate trainee on the Prospects website, I have attached my resume."
Perhaps the job description includes a Reference Number? It will be clear if it does, something like "Ref. No. -". If it does, include that both in this part, and also in the subject box of the email. It means a lot more to the employer than 'posting for a graduate trainee', so something like:
"In response to your job posting (Reference Number XXXXX) for a graduate..."
Student or Learner