Yes, you can move a Hunter's Mark while in Wild Shape.
That is, as long as you still have a Bonus Action left in that turn, and your character still holds their concentration on the spell. Hunter's Mark indeed mentions:
If the target drops to 0 hit points before this spell ends, you can use a bonus action on a subsequent turn of yours to mark a new creature.
Wild Shape says:
You can't cast spells, and your ability to speak or take any action that requires hands is limited to the capabilities of your beast form.
So you can't cast new spells, but on spells your character is already concentrating the rules continue:
Transforming doesn't break your Concentration on a spell you've already cast, however, or prevent you from taking actions that are part of a spell, such as Call Lightning, that you've already cast.
According to this highlighted rule, being a beast doesn't stop a druid/ranger from moving a mark, since Hunter's Mark was already active and the spell doesn't mention requiring hands to move it to a new target.
Let's also take a closer look at the example of Call Lightning:
On each of your turns until the spell ends, you can use your action to call down lightning in this way again, targeting the same point or a different one.
The wording of Call Lightning and Hunter's Mark are similar enough to conclude that yes, you can move a Hunter's Mark while in Wild Shape.