No. Run Plan is designed as a standalone experience for the Apple Watch. The watch handles interval guidance, haptic alerts (vibrations), and GPS tracking entirely on its own.
Note on Accuracy: While the app is built to be phone-free, carrying your iPhone with older Watch models (prior to Series 8 or Ultra) can improve GPS and pace accuracy. These older models often "tether" to the iPhone’s GPS to save battery and provide a more stable location lock. For Series 8, Ultra, and newer, the watch sensors are more than capable of pro-level precision on their own.