Making a stealth check can be done as part of the normal move. Check.
Shot on the Run lets you move up to your full speed and make a ranged attack during the movement. Check.
Hide in Plain Sight lets you make a stealth check without requiring cover or concealment, so long as you are near Dim Light. Check.
Making a stealth check while moving suffers a penalty of -5 when moving more than half but less than your full speed. Check.
So, with HiPS, and Shot on the Run: (assuming a speed of 30ft)
You should be able to be stealthed (within 10 ft of dim light), move <30ft, make a ranged attack dealing sneak attack damage (if any) within 30ft (barring abilities that could give more range), move the remainder of your original 30ft, and stealth again provided the 2nd part of the movement is within 10ft of Dim Light.
Just like a normal rogue with spring attack who is stealthed at the beginning of his turn can move in, take an attack with sneak added, and continue moving behind some bushes(which provide cover or concealment), giving him a new stealth check at the end of his turn.
