Bluebird Taxi's (the more reputable / trusted firm) has an app, and also recently launched booking via the GoJek / GoCar app (not sure if that alliance is available in Bali yet). read more
There *is* a taxi-booking app in Bali - for the best and only reliable and safe taxi company on the island - Blue Bird. The app works well, like others, for booking a taxi and directing the booking to the closest one to your current position. Works well. read more