Polymorphed creatures cannot activate any magic items
The rules on polymorph are very restrictive on magic item activation, this clause is not simply talking about your melded gear, but any activation magic item cannot be used on your form:
Items that require activation cannot be used while you maintain that form.
For instance, if you equipped the fox with tiny boots of speed, she could not activate them.
Even if the caster could activate the Caster's Tattoo, she could not cast while in the fox shape, as animals are not normally capable of spellcasting on their natural form as they lack ways to provide verbal, somatic and material components for spells. The first two are covered by the tattoo's magical effect:
A caster’s tattoo is infused with potential magical energy that aids in casting spells that normally utilize verbal and somatic components. Once per day, a spellcaster may activate her caster’s tattoo as a swift action to automatically enhance the next spell she casts in that round with both the Still Spell and Silent Spell feats.
You still require some form to remove material components while polymorphed, like using the Eschew Materials, which would allow you to cast any spell without an expensive material focus, as you are removing verbal, somatic and material components (or most spells).
Of course, im disregarding spells that lack some or all of those requirements, as they could be cast without special abilities that remove them.
It might not be possible to create this magical item on a kitsune turned into a fox, as the Fox Shape ability works as Beast Shape II, which lasts only 1 minute/level. You would have to shape in and out of this form all day for several days (minimum 3) before the tattoo is finished. But that is likely a memory lapse from the author of the feat, as Fox Shape is supposed to enhance your natural ability to change shape according to it's flavor. So you will see table variation here (ask your GM).
You can create magical tattoos, magic items inked directly into the flesh of a willing or helpless creature. Both you and the recipient of the tattoo (if the recipient is not yourself) must be present during the entire tattooing process.
Instead of a tattoo, you could also consider a Ring of Eloquence, which allows you to speak while on your fox form on four different languages. You still would require the Still Spell and Eschew Materials feats. That way, you will no longer be limited to a single spell per day and can cast any spells in your fox form. Though they use up slots 1 level higher if they have somatic components.