Due to modern roster restrictions (and a 2016 rule change discouraging the kicking of touchbacks by awarding the receiving team possession at the 25-yard line instead of the 20), most NFL teams do not elect to have such kickoff specialists, and instead use their placekickers (or, less often, punters) on kickoffs.
The most recent NFL pure kickoff specialist was Jordan Gay, who played the position for the Buffalo Bills under special teams coach Danny Crossman from 2014 to 2016.
Other players who have spent at least some time as kickoff specialist for an NFL team since 2006 include Stephen Hauschka, Rhys Lloyd, David Buehler, Todd Carter, and Brandon McManus.
As of the end of the 2014 NFL season, punters Thomas Morstead, Matt Bosher, Sam Martin, Pat McAfee, and Michael Koenen handle kickoffs.
Even college football teams usually do not implement dedicated kickoff specialists despite the much larger rosters at that level; however, if an underclassman has a stronger leg than the upperclassman kicker, but is not yet ready to assume placekicker or punter duties, they will usually handle kickoff duties while being the primary backup placekicker and punter.