Yes, if you're deficient in iodine or tyrosine your thyroid will not function properly. (t3 and t4 are actually kind of like 2 tyrosines containing iodine atoms), t3, 3 iodine atoms, and t4, you guessed it 4 iodine atoms) as you can see iodine and tyrosine are essential for production of both.
there is iodine in most multi vitamin/minerals.
and like huck said tyrosine, it's the combonation of both iodine and tyrosine in which is effective, because
iodine (iodinates) tyrosine so that the now iodinated tyrosine can under go coupling to create T3, not only that but t4 as well.