Yes, it's legal (...if it's the Druid holding the staff or club)
The description of infusing an item says:
Whenever you finish a long rest, you can touch a non-magical object and imbue it with one of your artificer infusions, turning it into a magic item.
It clearly points out that the object must be non-magical to start and becomes magical after the process. So the Artificer makes a magical staff.
Now the Druid comes along and casts Shillelagh:
For the duration, you can use your spellcasting ability instead of Strength for the attack and damage rolls of melee attacks using that weapon, and the weapon's damage die becomes a d8. The weapon also becomes magical, if it isn't already. The spell ends if you cast it again or if you let go of the weapon.
The spell calls out that the weapon can be magical. No magical "plus" bonuses to worry about stacking.
The spell changes the attack and damage rolls from Strength to Wisdom and updating the damage die. Nothing in the description would indicate it would cancel out the infusion; it doesn't care.
It also shows that the spell only lasts while the Druid is holding the weapon. Your phrasing didn't make it clear that the Druid would maintain control of the staff after infusion, so I wanted to point this out. I don't want you to assume that the Artificer could hold the staff and the Druid cast Shillelagh and it would still work.