ROME—When Italians vote on a much-awaited popular referendum on Sunday, they will also be deciding the fate of Prime Minister Matteo Renzi’s government—and expressing the country’s appetite for change.

The ballot is ostensibly over Mr. Renzi’s proposal to overhaul Italy’s legislature. But with his popularity waning and the economy stalled, it has become a make-or-break vote on the premier himself and his vision for a nimbler and faster-growing Italy. A loss would likely drive Mr. Renzi from office and usher in a period of...