The Fool: 'Om vajrasattva hum'
The mantra is chanted, traditionally, 108 times using a mala. I suggest using a crystal mala made of agate or tourmaline, or hold an agate or tourmaline in one hand as you chant. Or, you could chant it while holding your hands in the lotus mudra at the heart chakra, to symbolise purity and openness.
The Magician: 'Tattwam asi'
The mantra means 'I am that'. The intention of the mantra is to realise our oneness with the All, or to put it another way, to become in practice what we are in potential. To realise that we are that which we seek. The Magician becomes that which he seeks when he allows himself to be a lightning rod or conductor of the energy of the universe (of which he is actually a part) and the mundane world (of which he is also a part). He is the will unified and directed toward a goal, to create 'as above, so below'.
Try it holding a clear quartz crystal or labradorite, or a mala made of either while chanting this mantra. Or you could create your own style of 'as above so below' hand position (mudra) to accompany the chant.
The High Priestess: 'Om mani padme hum'
This well-known mantra means 'Hail, jewel in the lotus,' or 'The jewel of consciousness is in the heart's lotus'. The phrase suggests hidden depths, as does the High Priestess. She represents dark mysteries, psychic forces, a wisdom that cannot be expressed in rational terms. And neither can the meaning of this mantra, which carries far more meaning than its surface suggests. Try holding a moonstone when chanting this mantra and contemplating the High Priestess.
These are just my own associations, based on nothing but the connections that I personally have made between mantra and tarot for my own practice.