It is unlocked and upgraded instantly as long as you have the mats.

Supplements are available from a new game mode SSS, which will run in two phases monthly(reset on 1st and 16th of each month). Each phase will let you get a maximum of 30x T3 supplements and 12x T4 supplements. So every month you can get upto 60x T3 supplements and 24x T4 supplements from that game mode. A 6* requires 60x T3 supplements for Mod2 and 20x T4 supplements for Mod3 so you can completely upgrade one 6* module every month.

Additionally, there are one-time missions which will give roughly enough supplements to raise two more 6*. And the remaining supply is through events but it is a trickle.