How to add a sprinting mechanic to my game?

Phaser 3
1

I want to add a feature to my game where if Shift is held down, the player will move at an increased velocity.

2

That should be fairly straight forward due to the way game-loop logic works. Each call to the update method you are moving your character if they are pressing a direction. If you take this example on moving a character with the keyboard, you can add a run mechanic with shift by modifying the update() logic to this:

function update ()
{
    player.setVelocity(0);
    let velX = 0;
    let velY = 0;

    if (cursors.left.isDown)
    {
        velX = -300;
    }
    else if (cursors.right.isDown)
    {
        velX = 300;
    }

    if (cursors.up.isDown)
    {
        velY = -300;
    }
    else if (cursors.down.isDown)
    {
        velY = 300;
    }

    if (cursors.shift.isDown) {
        velX *= 2.5;
        velY *= 2.5;
    }

    player.setVelocity(velX, velY);
}

You can see I am just checking for the Shift key being held down, and if it is, I increase the velocity the box is moving at