No, she is not a royal (as she is not a 'royal highness') but she is still a member of the British royal family. And she, like anybody else, is free to wear a tiara whenever she likes; and especially at her wedding day. It would have been more surprising if the daughter of the princess royal would not have worn a tiara on her wedding.
N.B. Technically no titles for Zara and Peter were refused. Anne and Mark weren't interested in a title for Mark which had the consequence that their children don't have a style. Had their father been made an earl, Peter would have been Lord Peter (or more likely Viscount X - assuming that Mark had also received a subsidiary title like Anthony did) and Zara Lady Zara. Their half-sister Stephanie would be Lady Stephanie. Their other half-sister, Felicity, would have remained 'plain' Felicity as she was born out of wedlock (while Mark was married to Anne).