Sometimes yes, sometimes no.
They usually make wind for the Harriers, because they have a narrower wind envelope and it adds a margin to their generally tight takeoff and landing power margins.
For everyone else, it depends. They will if they can, but often the only make steerage, maybe 3-4 knots. Sometimes they're "on the hook," (when they party though, they're "off the hook") or at anchor during flight ops.
When the ARG is on the move, often they're moving too fast for winds or in the wrong direction. The air plan can get complicated then, because they'll plan to go off course just long enough to get winds for flight ops, then resume course. They steer for winds again when one lands.
Well deck ops usually work around flight ops, vice the reverse, unless there's a big surface offload. The ship turns to accommodate the LCAC, pausing flight ops, then resumes once the well deck is done with recovering/deploying.
