Rule 902.5a

If a player takes a mulligan in a Vanguard game, just like in a normal game, that player shuffles their hand back into their library, then draws a new hand equal to their starting hand size. (In a multiplayer game, a player’s first mulligan is for the same number of cards as they had before.) See rule 103.5. Example: The hand modifier of a player’s vanguard card is +2. That player starts the game with a hand of 9 cards. If the player takes a mulligan three times before keeping, they draw a new hand of nine cards and put three of those on the bottom of their library.