I've applied for a job, been invited for an interview but was unsuccessful.

I did request feedback, but they haven't responded.

Is it a legal requirement that they do respond?
Will I be able to something about it?

Sorry to say but there is NO legal requirement for employers to give feedback to unsuccessful interviewees.
To answer your question in order, No and NO.

I agree with Curlyben. No and No. This applies to an interview with a new company and also within your own company (if your transitioning to another position in-house).

But... if this interview was in your current company, you already have the feedback. Why? If the interviewer does not want to take the short amount of time to validate your feelings, you're probably better off not getting that job in that dept under that manager... Who'd want to work with someone who doesn't set good goals, and truly cares enough about the people within his/her team to help them grow and improve?

One note: Post-interviews are only as good as letting you know you didn't make the job, and generalizations on "work on your answers." You won't get specific feedback on style, poise, or any of your answers. This is because if the interviewer is too specific, it sets a precedence and possible advantage if you go another time for an interview on that job and are denied. If you did everything that the manager says you didn't do well, and you nail those issues in the next interview, they really don't have a reason to deny you.

I believe you did as well as you could in the interview. Good Job! Now ask some questions yourself? Did you have all the educational requirements? Time in job requirements? If you're denied a job, what about the culture of the department? Do you fit in? Do you know if you'd fit in? Do you think you'd fit in to the team? Try to understand from these three standpoints... WILL you do the job? CAN you do the job? Do you FIT the job?

"for every no we're one step closer to a yes!" Good luck on the job search.