In this great answer I learned that the Mars rover Curiosity can be given some tasks and it will go ahead and manage the work and navigation by itself, to at least a certain limit.
The James Webb Space Telescope needs to maintain its attitude so that the giant mirror and other optics and experiments stay dark and cold - about 50 Kelvin, while its solar panels get enough sunlight to produce about 2 kilowatts of electricity.
It will also be in a somewhat unstable halo orbit around Sun-Earth L2. Typically spacecraft are monitored from Earth stations, and doppler measurements are precise enough to decide when orbit correcting thrust maneuvers are necessary to keep it from wandering away very much. Without careful, regular small corrections, much larger fuel-gobbling corrections would be needed.
These corrections are usual surprisingly small, of the order of a kilometer, and deep sub-meter-per-second in velocity. Precise doppler measurements are combined with detailed orbit modeling including perturbations are used to calculate each maneuver on earth, then checked and double-checked before the instructions are sent to a spacecraft in a halo orbit. Afterwards a second set of measurements are made to confirm that the maneuver had the intended correcting effect on the orbit.
Could the JWST manage its own regular orbit correction maneuvers without help from Earth if absolutely necessary? Suppose there's a failure or a meteorite hit that caused a communications problem - can it at least maintain an orbit until a repair mission (crewed or automated) comes to fix it?
If so, how can it measure its own position precisely enough so that the regular corrections are small? While star cams can be used to ascertain attitude, how will position be measured precisely? Are occasional GPS measurements possible at that distance? Does it have good enough visible light cameras to make super-precision terminator measurements or even pattern recognition of landscape features or occultation timings of/by the Earth and the Moon?