The meaning of "You" should be consistent
Delivering touch spells via a familiar is usually a simple matter:
when you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell. (PHB, p. 240)
Usually, this means the spell functions as though you had cast it (for purposes of effects and modifiers), except that the familiar defines the range of the spell (within its reach).
But Shillelagh has a couple of terms within it that make the question of who had cast the spell problematic. Specifically:
for the duration, you can use your spellcasting ability instead of Strength for the attack and damage rolls of melee attacks using that weapon... The spell ends ... if you let go of the weapon.
Now, the meaning of "you" in the above sentence should be consistent. Either "you" is always the spellcaster or "you" is always the familiar. But neither option is good.
If "You" is the spellcaster:
In this case, the spell would automatically end immediately, because you are not holding the weapon in question, and thus you have effectively "let go of the weapon."
If "You" is the familiar:
As Szega pointed out, the familiar has no "spellcasting ability," and thus the damage would default to strength anyway (it would become a 1d8-4 attack and magical, but gain no other benefit). In many scenarios, this would be worse than its basic attacks. Or (as David Coffron pointed out in a comment) a DM might rule that the rules on magic items apply, and:
If you don't have a spellcasting ability...your spellcasting ability modifier is +0 for the item, and your proficiency bonus does apply. (DMG, p. 141)
Although this would become a 1d8+0 (a significant upgrade from the 1d1 it usually gets), it wouldn't be as effective as if you cast the spell on yourself. And the Sprite's poisoned shortbow would often be more effective (and certainly more likely to hit with a +6, rather than the +2 it would get from Shillelagh).
A DM could rule otherwise if they so desired (making "you" mean two different things, or ruling that since you never technically held the club in the first place, you haven't technically "let go" of it). But as written, the rules tend to discourage this strategy.