Like LabRat, I know as little as possible about the monarchy.
I do know, however, that because the monarch is also head of state there are political implications for who she will meet. Therefore no-one would be able to gain an audience with the monarch without political vetting.
So, let's say that your character had recently expressed some controversial views. Just for the sake of an example, let's say your character had been critical of Western governments for selling arms to certain countries in the Middle East. Were the monarch to meet your character, that could be spun in the media as the queen indicating approval for those views. (We all know how ridiculous that notion is, but it's the way these things get spun).
So what I'm getting around to saying is that one key route to getting an audience would be through political, not aristocratic, contacts. So, if you're talking about an American politician, then one good route is through the British government - is your politician friendly with any Cabinet ministers? Got good contacts at senior Civil Service level?
And I think another good route would have to be via the American Embassy in London (assuming your character is American). All ambassadors to the UK are accredited to the Court of St James, making the post one directed at the monarchy and not at the British government. Just as in the US the President receives new ambassadors' credentials, in the UK it's the monarch who does so. Ambassadors get invited to formal receptions at Buckingham Palace and move in royal circles as appropriate as part of their job.
So I'd go with government and embassy contacts.
Wendy