Since you are talking about "accost" as a word rather than using it for its meaning, it is necessary that you place it within quotes. It is just a wonderful word, no more or less formal than any other. Why would you want to replace it?

You could use ’approach’. It is quite common in some places in London for people employed by charities to try to talk to people in the street in an attempt to persuade them to donate money. It can be ’annoying' as Lauren suggests, but the intention of the people who go up to strangers is not to annoy them.

I would say depends on the context a little bit but, ones I tend to frequently use myself is ‘bother’, or ‘annoy’. I kept annoying them until they responsed. The paparazzi kept bothering the actor at lunch.