When a feature requires a free hand, there will be something in the rules to tell you that.
We know that casting a spell with somatic components requires a free hand because there is a rule for that:
If a spell requires a somatic component, the caster must have free use of at least one hand to perform these gestures.
We know that picking up and attacking with a greatsword requires two free hands because there is a rule that tells us that:
Two-Handed. This weapon requires two hands when you attack with it. This property is relevant only when you attack with the weapon, not when you simply hold it.
There is no rule anywhere that tells us that the Archer option of the Star Druid's Starry Form feature requires a free hand. So it doesn't. We often see the phrase used in discussions about the rules, "there are no secret rules". What this means is that there is no rule governing the function of your class features that you do not have access to. Everything you need to know for this class feature is written right there in the feature description.