You can grapple and move a creature you're frightened of.
The Cause Fear spell simply causes the frightened condition and has no other effect:
The target must succeed on a Wisdom saving throw or become Frightened of you. The frightened target can repeat the saving throw at the end of each of its turns.
The Frightened condition says:
The creature can’t willingly move closer to the source of its fear.
But moving a creature you are grappling does not require you to move closer to the grappled creature. You're already as close as you can possibly get, in that you're literally touching.
So even when frightened, moving a grappled creature functions as it normally would:
When you move, you can drag or carry the Grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
However, you'd have disadvantage on the Strength check to maintain the grapple if the source of fear tried to escape.
More powerful effects, like that of the Fear spell, place additional requirements on the frightened creature:
While frightened by this spell, a creature must take the Dash action and move away from you by the safest available route on each of its turns, unless there is nowhere to move.
But the Frightened condition by itself (as caused by the Cause Fear spell, Wrathful Smite, or a dragon's Frightful Presence) does not have this more powerful effect.