Given that it's not a move that requires a roll, I would think yes. Thoughts?
The answer can be extrapolated from the wording of the Shapeshifter move:
... You have any innate abilities and weaknesses of the form.... You still use your normal stats but some moves may be harder to trigger—a housecat will find it hard to do battle with an ogre.
Since the species you are shapeshifted into has the ability to interpret the "grunts, barks, chirps, and calls" mentioned in the Spirit Tongue move, it makes sense that you would be able to do so while shapeshifted. It would not make sense for a housecat to have a hard time understanding others of its kind.
Can the shapeshifted druid use Spirit Tongue to understand the communication of species other than his or her current form? That comes down to the fiction and the kind of world being portrayed. Is it a world where all animals can speak with each other, like a Disney movie? Or is it a world where each species has its own language? Either one can be interesting. If you don't know or can't decide, ask the druid and use their answer, following the GM Principles.