That trio completed three rounds at 13-under-par 203 and have seven more players within three shots of their lead. Jon Mills, Kyle Thompson and Esteban Toledo are one stroke further back at The Country Club at Woodmore.
Claxton dropped a shot at the second, but six birdies the rest of the way gave him a share of the lead.
Van Zyl was just 1 under for his round through 14 holes. To that point, he had a double-bogey, two bogeys and five birdies. Van Zyl birdied the final three holes to finish at 13 under.
Sheehan collected five birdies over the first 15 holes, but a bogey at the 17th cost him the outright lead.
James Driscoll is alone in seventh place at 11-under-par 205 after a third- round 68. Chris Thompson, Marc Turnesa and Matt Weibring are tied for eighth at 10-under-par 206.