The Officer on Duty is responsible for picking up the league-provided game balls (2) from the Press Box or Concession Stand prior to each game and delivering them to the fields.
At the end of the game, each team is given one game ball. It is an ILL tradition (especially in the younger divisions) that the Managers and coaches of each team choose one player from their team to receive that day's game ball.
The Team Parent may want to keep up with who gets the game ball each game (usually note it in the scorebook) so that all players have an opportunity to earn the game ball throughout the season. The Manager will typically write the date and player's name on the ball. (One unique way to memorialize that game ball for each player is to choose "game ball trophies" for your end of the year mementos - these are trophies designed to hold a game ball - and they can be found at various online retailers).