The Republican Party’s best-connected political operatives have quietly built a massive fundraising, organizing and advertising machine based on the model assembled by Democrats early in the decade, and with the same ambitious goal — to recapture Congress and the White House. The new groups could give Republicans and their allies a powerful campaign apparatus separate from the Republican National Committee. Karl Rove, political architect of the Bush presidency, and Ed Gillespie, former Republican Party chairman, are the most prominent forces behind what is, in effect, a network of five overlapping groups, three of which were started in the past few...