According to the JWST FAQ:
To turn and point at different objects in space, Webb uses six reaction wheels to rotate the observatory. The reaction wheels are basically flywheels, which store angular momentum. The effect of angular momentum is familiar in bicycle riding. It is much easier to stay up on the bike when it is moving than when it is standing still, and the bicycle will tend to go straight in 'no hands' mode thanks to the angular momentum of the spinning wheels. Slowing down or speeding up one or more of the Webb's reaction wheels alters the total angular momentum of the whole observatory and consequently the observatory turns to conserve angular momentum. Hubble uses reaction wheels also to turn to point at different objects.
The reaction wheels work in combination with three star trackers and six gyroscopes that provide feedback on where the observatory is pointing and how fast it is turning. This enables coarse pointing sufficient to keep the solar array pointed at the Sun and the high-gain antenna pointed at the Earth. To take images and spectra of astronomical targets (i.e., galaxy, star, planet, etc.) finer pointing is needed. Additional information for finer pointing from the Fine Guidance Sensor in Webb's integrated science instrument module (ISIM) is used to move the telescope's fine steering mirror (FSM) to steady the beam of light coming from the telescope and going into the science instruments. Webb's reaction wheels, star trackers, gyroscopes, Fine Guidance Sensor, and fine steering mirror work together in the observatory's attitude control system (ACS) to precisely point and stare at targets so that the science instruments can see them and see them clearly. The system works much the same way your body uses multiple methods of differing precision ("your inner ears and eyes and nervous system and muscles") to catch a baseball in the outfield.
To me, this implies the JWST's mirrors are rigidly attached to the base of the spacecraft, so they're in a fixed orientation. The telescope always looks in a plane roughly parallel to the sunshield.
Because the sunshield has to be normal to the Sun (more or less), that limits the field of view available at any one time to a circular strip of the sky. If you wanted to look at a particular star, you'd have to wait for up to 6 months for it to become available. That seems too long for many interesting transient phenomena.
The key here is the 'more or less' part: how far out of the normal can you tilt the JWST without losing the sunshield's effectiveness? How much of the sky can the JWST observe from any given position in its orbit?