We investigate Wolf-Rayet (WR) stars as a source of feedback contributing to the removal of natal material in the early evolution of massive star clusters. Despite previous work suggesting that massive star clusters clear out their natal material before the massive stars evolve into the WR phase, WR stars have been detected in several emerging massive star clusters. These detections suggest that the timescale for clusters to emerge can be at least as long as the time required to produce WR stars (a few million years), and could also indicate that WR stars may be providing the tipping point in the combined feedback processes that drive a massive star cluster to emerge. We explore the potential overlap between the emerging phase and the WR phase with an observational survey to search for WR stars in emerging massive star clusters hosting WR stars. We select candidate emerging massive star clusters from known radio continuum sources with thermal emission and obtain optical spectra with the 4m Mayall Telescope at Kitt Peak National Observatory and the 6.5m MMT. We identify 21 sources with significantly detected WR signatures, which we term emerging WR clusters. WR features are detected in $sim$50% of the radio-selected sample, and thus we find that WR stars are commonly present in massive star clusters currently emerging. The observed extinctions and ages suggest that clusters without WR detections remain embedded for longer periods of time, and may indicate that WR stars can aid, and therefore accelerate, the emergence process.