Capacitor bank for high powered appliance
WARNING

AC MAINS WILL KILL YOU WITHOUT QUESTION IF YOU LET IT.

DON'T LET IT!!!

THINK before you "do"
Draw out any circuits with switches shown open or closed.
Work out what will be connected where and why in each case.
Do not expose yourself to any situation that MIGHT kill or injure you worst case. 

So:



This occurrence is unusual.
Is the circuit breaker correctly rated?
Is the circuit a low current one not suited to this load level?
Is the breaker faulty - some breakers that have seen long service suffer a decline in trip current - try another with the same rating? What happens.

The vacuum cleaner MAY be faulty.
What is its nameplate voltage and current?
What is the fuse or breaker current rating?

Then:



Capacitors will not work on AC 

and

Energy content needed is large compared to what caps can provide even if you could convert the stored energy to AC.

A better method which will almost certainly do what you want is to provide a limiting series resistor - a large wattage light-bulb or a part of an old toaster or heater element.  You need enough resistance to limit the current but not enough to present the motor running up at all or in a reasonable time.

A vacuum cleaner usually uses a "series wound" motor and will tolerate series resistance and will run at lower speeds at lower voltages. A few modern vacuum cleaner motors are rated at 2 kW, but most will be not much over 1 kW at most and many are lower again - maybe 500W. 

LIGHTBULB: A 100 W incandescent lightbulb will present more like a 1 kW load when cold and may make the difference. As it starts to heat the resistance rises and vacuum motor voltage will fall so a bulb will only be useful for a brief burst.

HEATING ELEMENT: You can easily trial an old heater or toaster or kettle element or elements or use the wire from ones which have open elements.

Place an element in series with the vacuum cleaner with a switch across the heater element which is open circuit to start and closed circuit to run. Several elements in parallel will provide a higher starting current. Wire stripped off a toaster or heater element "should" withstand mains current levels liable to be experienced in this application.