The description of a Spellwrought Tattoo states:
Produced by a special needle, this magic tattoo contains a single 1st level spell, wrought on your skin by a magic needle. To use the tattoo, you must hold the needle against your skin and speak the command word. The needle turns into ink that becomes the tattoo, which appears on the skin in whatever design you like. Once the tattoo is there, you can cast its spell, requiring no material components. The tattoo glows faintly while you cast the spell and for the spell's duration. Once the spell ends, the tattoo vanishes from your skin.
The Ability modifier for this spell is +3; the Save DC is 13 and the attack bonus is +5.
The two bold parts are the parts I'm wondering. The first part implies that you cannot apply it on someone else since it says you must hold it against your skin, and the second half of that sentence implies you need to actually be able to speak in some fashion. So if you have a familiar such as say, a Crow, could it say a command word through it's mimicry? What about a creature such as an owl or cat? Could the command word be a bestial noise?
But let's assume that instead of a standard familiar, you're using a unique one that is already capable of verbal communication such as an Imp. While a familiar cannot attack, it can "take other actions as normal." Could it then use the spellwrought tattoo to cast whatever spell is on it, such as Magic Missile, Burning Hand, Silvery Barbs, or Shield of Faith? What about spells such as Find Familiar, since you are not casting the spell, but your familiar is.