What pressure will your vacuum pump pump to? You don't really want a traditional vacuum pump, we are dealing with input pressures of up to 2100psi (in this example) and output pressures from 3000-4000psi.
If you are asking questions like this then you probably are more than one weekend away from pulling it off safely. You need to grasp the relationship between pressure and volume, and how the gases equalize when connected.
Without a proper compressor or booster you won't get more than 2100 psi from the extra tank, with the amount falling a lot on subsequent dives.
The first dive, lets say you take the tank on your back to 100psi. You hook it up to the storage bottle you rented, and open all the valves on your whip, and two tanks. Gas moves back and forth until both bottles are at the same pressure. Lets say that pressure is 1900psi. You'll have taken about 60 cubic feet of gas out of the storage tank and put it in your scuba tank.
Next dive, you do the same thing, but now they equalize at 1600psi and you only put 45 cubic feet in your tank.
Third dive they are at 1200psi and you only put 30 cubic feet in your tank.
Of course those numbers are just wild guesses, but they illustrate the concept.
With a booster (a compressor which can accept an input pressure significantly higher than ambient and go all the way to 0) you can use the storage bottle at a lower pressure and fill the scuba tank to the proper fill. However, you will also need a third drive gas at a lower pressure but at a very high volume, to run the booster.
It might be cheaper to buy a small portable compressor, but then you have to educate yourself on not only the equalization stuff, but also filters and maintenance. For most "casual" divers it's simplest to buy a few extra scuba tanks. Also, consider the amount of space and weight that the fill setup will take up.
Remember, pressurized gases contain immense amounts of potential energy, and breathing gas impurities can be fatal. It's not rocket science, but it is very important to be careful, safe, and understand what you are doing.