BELGRADE, Serbia—Europe is distracted by internal discord over immigration and its tense relationship with Russia and the U.S. Seeking to fill the void, China is taking advantage of a historic opportunity to wedge itself into the heart of the West.

Deal by deal, applying experience honed in Asia and Africa, China is constructing parallel financial and commercial networks in Central and Eastern Europe to challenge the global order. It has taken footholds in more than a dozen nations on the periphery of the European Union. Some,...