Camp has appeared in many television series and received a Primetime Emmy Award nomination for his role in the miniseries The Night Of (2016).

In 2012, Camp joined Philip Seymour Hoffman and Andrew Garfield in Mike Nichols' Broadway revival of Death of a Salesman at the Ethel Barrymore Theatre.

In 2016, he starred in the Broadway revival of The Crucible alongside Saoirse Ronan, Ben Whishaw, Ciaran Hinds and Sophie Okenedo at the Walter Kerr Theatre.

Off-Broadway credits include starring as Gordon in Sarah Ruhl's Dead Man's Cell Phone at Playwrights Horizons in 2008, before having to withdraw due to other work commitments.

On August 30, 2021, it was announced that Camp would appear as Matthew Burke in an adaptation of Stephen King's 'Salem's Lot for Warner Bros. Pictures and New Line Cinema.

Camp has also appeared in Law & Order, The Good Wife, The Leftovers and The Night Of, receiving a Primetime Emmy Award nomination for the latter.