Hopefully, you weren't thinking of putting the relay activating terminals in series with the drill power. There are two ways to do what you want. One is to have an external, switched duplex outlet that supplies power to both the drill and vacuum. The switch should be double pole and rated for the drill plus vacuum HP. This is the simplest and cheapest way, but may be inconvenient in some cases. As another posted, if you want to use a relay, you'll need more than just the relay. You'll need a current sensing circuit that detects when power is going to the drill and that is what should activate the relay. The relay should be double pole, have the correct voltage rating for activation and contacts, and be rated for the drill motor load (i.e. have a HP rating). What you really need is a motor contactor, which is a relay designed specifically to control motors. If you buy a premade sensor/controller, be sure it's rated for your drill and vacuum HP.
You can't just take the current rating of a relay and assume it will be adequate for motors. They are inductive beasts with big starting and stopping surges in most cases. The relay needs to be rated for a motor load.
Vacuum HP is often grossly overstated on the label; you'll need to read the fine print in the manual or on the motor. I have shop vacuums rated to operate on 15A, 120 VAC circuits, and the labels on the units state they are 3 HP or more! The manufacturers lie by converting the starting surge current or locked rotor current to HP and use that in their ads.
Mike in California