I'm not a rules expert by any means, but would that work to get TWO taps from the same Minister to pay for the ability in between upkeep and draw steps?
502.3. No player receives priority during the untap step, so no spells can be cast or resolve and no abilities can be activated or resolve. Any ability that triggers during this step will be held until the next time a player would receive priority, which is usually during the upkeep step. (See rule 503, “Upkeep Step.”)
At the beginning of the upkeep step, if you place the bounce trigger first on the stack from Trusted Advisor targeting Minister, then put Persistent Petitioners ability on the stack; you TAP Minister, Petitioners, Advisor, and you still need one other "advisor card" to pay the cost to resolve the ability. To me it looks like you need one more "advisor card" on the board (so 4 cards minimum are needed to play the ability of Petitioners during upkeep).
I'll reiterate a likely setup that's within reason.
T1: Mutavault, play Aether Vial
T2: Island, play a 1CMC Minister from Vial in main phase or opponents next end step, cast Persistent Petitioners [2 total "advisors" on battlefield plus mutavault] fully tapped out for mana though
T3: UNTAP, play 1CMC Advisor from Vial in upkeep (tick Vial up to 2 on stack first). Activate Mutavault and tap all creatures to play Petitioners ability. Mill 12 with 3 "advisor" cards plus Mutavault during upkeep. DRAW step. Continue play.
T4: UNTAP, Trusted Advisor trigger on stack targeting Minister, activate Petitioners ability (tapping all the same advisors and mutavault from T3), Vial in a 2CMC "advisor" if you want or Manic Scribe to be target for future bounce back. Resolve stack to play 2CMC "advisor"/Manic Scribe if you did (mill 3), mill 12, bounce back Minister. You milled 24 or 27 by T4 DRAW step and have 1 or 2 Persistent Petitioners, 1 Trusted Advisor, and 1 Mutavault at least in play now and probably 2 untapped lands if you didn't miss a land drop.
Again, please correct me if I'm not resolving these cards correctly in Modern based deck. I would think you would not need more than 8 Petitioners, plus draw like Visions of Beyond and good mill Archive Trap and Glimpse the Unthinkable. By turn 4, half of your opponents deck will be in the graveyard and likely scooped unless it feeds into their strategy.