The run is critically important! Regardless of size and weight, an athlete can increase their vaulting ability by running faster, especially the last few steps before the board...
...because, physics.
Increased velocity causes increased weight upon impact. A 60 lb. girl hitting the spring board at 10 miles per hour will weigh more, and thus, deflect the springboard more, than a girl who impacts at 8 miles per hour. Yes, bigger girls may be at an advantage, but by learning to truly "unleash" their full power in those last few steps, smaller girls can really get a great punch from the board. Think about it this way: if someone lobs a baseball to you and you catch it, it's no big deal. However, if a major league pitcher threw a fast ball at you, it would make your arm recoil back when you caught it. Due to the increased speed, the ball has more weight at the moment you catch it.
Also, as dunno has said, there are lots of technique factors involved after the run that make a huge difference, too. The entire vault must have an overall rhythm that doesn't slow down at any phase. Then there are other factors, like the amount of springs in the board, the distance of the board to the table, the height of the table, the technique of the gymnast, and the actual vault being performed. Vault is a complicated event, and all for something that is over in about eight seconds.
All that being said, the run is the very first phase of every vault. It absolutely must be refined and perfected if any success is to be had.