I've been to Bishop many times and don't recall ever seeing a taxi there. It's a small town (pop 4000).
The whole town is only about a mile from edge to edge, so presumably people who don't have cars walk. It's hard to imagine that any significant market would exist for taxi service inside the town. Where are you trying to go?
This is the western desert region of the US. Distances are large, the population is spread thinly, and people drive cars to get around.
Eastern Sierra Transit is a long-distance bus line that runs up and down 395. For instance, if you want to go northbound, you have to go on Mon, Wed, or Fri.
[EDIT] Based on your comment explaining what you're trying to do.
Some options: (1) High Sierra Shuttle, cell 760.920.8266, office 760.872.3950. (2) Rent a car. (3) Approach a local and offer to pay for a ride.
If you don't rent a car, how are you going to get back from the trailhead?
The White Mountains are not heavily traveled, so I doubt that hitchhiking is going to work.
I don't know about the specific trailhead you're using, but in general roads to trailheads in this region can be of poor quality. Think twice about driving a low-clearance rental car to the trailhead, unless it's paved the whole way.
The following is off topic, but ... I think you posted earlier about hazards from wildlife in this area. Are you going in summer? If so, then sun exposure, overheating, and running out of water are bigger hazards to worry about. Basically you're hiking into a high-altitude desert, although there may be canyons with creeks in them. Note that the existence of a creek on a topo map doesn't necessarily indicate that the creek is flowing at any given time. This has been an unusually dry year in the Sierra. Your cell phone is almost certain not to work. Do you have a Spot or PLB (personal locator beacon)? Are you hiking with other people who are experienced in this environment?