When the mount and rider are charging into combat, and the rider is hit with an banishing arrow, what may happen?
The rider is banished if they fail the save.
Does the mount keep moving without the rider?
Worgs are intelligent mounts (Int 7), so you're dealing with a creature that acts independently per the rules for mounted combat:
Intelligent creatures, such as dragons, act independently...
An independent mount retains its place in the initiative order. Bearing a rider puts no restrictions on the actions the mount can take, and it moves and acts as it wishes. It might flee from combat, rush to attack and devour a badly injured foe, or otherwise act against your wishes.
Does it get its own separate turn while the rider is banished?
Per the rules for independent mounts above, it always has its own separate turn.
When the effect ends, will the rider appear back in the space where they vanished? Or will they reappear back on the mount some distance away?
When they come back, they'll reappear in the space they left or the nearest unoccupied space, per the rules for that feature. Whether they fall on their duff or not would be a DM ruling; however, the Mounted Combat rules I cited above have some useful guidance:
If your mount is knocked prone, you can use your reaction to dismount it as it falls and land on your feet. Otherwise, you are dismounted and fall prone in a space within 5 feet it.
This DM would force the Hobgoblin to burn their reaction to not fall prone.
As an aside, canonically worgs are incredibly vicious and dangerous intelligent mounts that are only slightly less likely to kill their handler than to kill their enemies. So once a worg has elected to charge the party, don't expect the banishment of their rider to deter their attack.