Utah Transit Authority buses run all over Provo and have a couple of stops along BYU's campus; they're also an effective way to get to Salt Lake. Most big apartment complexes have a bus stop right in front of them, and the buses run fairly regularly. While this is not a perfect solution to the lack of parking at BYU, it does help.
As for getting around in the city, it is possible to survive without a car, though not enjoyable. Provo is fairly spread out, and not everything is accessible by bus. There is no metro or train system to augment the bus system. Salt Lake is a bit better with the TRAX system, but Utah still has much to learn about efficient public transportation. To check out options, see the Utah Transit Authority Web site at www.utabus.com.