Yes, you can visit the Pro Shop during your round to replenish the ball
you are using. If you lose your last ball of a specific type and color
you can purchase new ones in the Pro Shop but will not be able to use
them in your current round if you have already switched to the Basic
ball.
You can buy balls during a round but regardless of whether or not the ball was changed during the round you can't use the new purchase until you equip your bag and that won't happen without navigating away from the game. Been there :(