Tapper was born in Bromley, South East London. She trained at the Academy Drama School and the Central School of Speech and Drama, from which she graduated in the spring of 2003, days before taking on her first film role.
Her television credits include Mary Collins in A Harlot's Progress for Channel 4, Jane in Oliver Parker's The Private Life of Samuel Pepys, Gemma in the first series of the Sky One drama Hex and Jenny Maple in the BBC miniseries 20,000 Streets Under the Sky. She also played Hermia in ShakespeaRe-Told: A Midsummer Night's Dream, the 2005 BBC adaptation/modernisation of Shakespeare's play of the same name.
In 2008 Tapper portrayed Sheila Steafel in the BBC television play The Curse of Steptoe and Anya Raczynski in the BBC remake of Survivors, alongside Max Beesley, Paterson Joseph and Julie Graham. She also played Selina Dawes in the ITV adaptation of the novel Affinity, opposite Anna Madeley as Margaret Prior. It premiered at the Miami Gay & Lesbian Film Festival, and in the UK on 28 December 2008 on ITV.
In 2009 she appeared in the ITV fantasy drama series Demons as blind vampire-turned-monster hunter Mina Harker. She played Effie Gray in the BBC Two period drama Desperate Romantics. In 2010, she played Hannah in the BBC television pilot Reunited.
In 2013 Tapper starred as Ellen Love in Mr Selfridge.
