The ambitions of Takeda Pharmaceutical Co. 4502.TO -0.81% go well beyond its $14 billion cross-border acquisition of Nycomed: Japan’s biggest drug maker now wants to rub shoulders with the likes of GlaxoSmithKline GSK.LN -0.11% PLC and Sanofi SA SAN.FR +1.48% as a global leader in the lucrative market for vaccines. Fueled by demand from emerging countries like China and India, the vaccines business is set to grow by more than half to at least $30 billion over the next five years, market watchers say. This year’s new global push by Takeda, still ranked only 12th by sales among drug makers world-wide, comes as industry players reassess the appeal of vaccines amid a flood of generic products that has cut into the profits of blockbuster treatments. Though costly and time-consuming to develop, vaccines are nearly impossible to copy in generic form. Analysts believe it is too early to make a call on the scale of Takeda’s vaccines potential from its current base of just under $300 million in annual revenue. But the firm has surprised even skeptics with the speed with which it has been planting its seeds, luring talent from overseas and hunting aggressively for technologies that will lift it into the top ranks.