A Jesuit high school in Manhattan’s Chelsea neighborhood that shares a beat-up Astro-turf field with dozens of youth soccer teams makes for an unlikely nerve center for the future of rugby in America. But it will have to do for now.

The 2015 Rugby World Cup gets underway Friday in England, and for the first time, the USA Eagles are realistically thinking about being something more than international cannon fodder. No one expects the Eagles, who open against Samoa Sunday, to challenge traditional powerhouses like New Zealand,...