Can a vacuum cleaner create sufficient vacuum to operate the artificial horizon?
The answer is a qualified yes: It's theoretically possible, but wholly impractical to provide a pressure source to your gyros from within the cockpit, at least the way you're describing.



Air-Driven gyros require a relatively constant pressure differential somewhere between 4 and 6 inches of mercury to operate correctly (this can be "suction" or "pressure", but as vacuum/suction systems are more common let's stick with that term). We can split the difference and say we want to supply 5 inches of vacuum (which is usually right in the middle of the "green arc" for most vacuum gauges) - this is sufficient suction to keep the rotor spinning at an appropriate RPM so the instruments function as expected.

Because a constant vacuum is required we can kill off one of the options: You won't be able to suck air through a straw to do this -- you would be constantly over-speeding or under-speeding the gyro rotor as you produce either too much vacuum (inhaling) or not enough (when you have to let go of the straw to exhale).
The variable pressure would vary the rotor speed, and your instruments would be totally unreliable within a few breaths, defeating the purpose of the exercise.

A household vacuum can provide the necessary pressure differential (these fine vacuum-rating folks tell me that "An average vacuum will have a [suction] rating of around 80 inches of water", which Google tells me is around 5.9 inches of mercury. If you could hook the vacuum up to the aircraft's system downstream of the vacuum regulator (where the main vacuum pump hooks up) you could theoretically power your gyros normally.
There are some caveats to using a vacuum cleaner idea besides the obvious "How are you going to hook the vacuum up to the system?" question:


The vacuum cleaner specifications are presumably at or near sea level - they might not be able to provide adequate suction at higher altitudes. 
Powering the vacuum cleaner in this scenario is left as an exercise for the reader (most aircraft with air-driven gyros don't have household AC power in the cockpit - you could install an inverter, but vacuums pull quite a lot of power, so you'd need a high-amperage circuit to supply it. The draw from the vacuum may exceed the capacity of the plane's electrical system, and to remedy that we're getting into complicated major alterations...)




There are other more practical alternatives to an engine-driven vacuum pump to consider if you're worried about the reliability of your vacuum source:


Precise Flight makes a standby vacuum system that ties into the engine intake.
For normally-aspirated engines this works well as long as the engine is operating at part throttle: The manifold pressure differential is sufficient that it can supply suction for your gyros.  The downside is that you can't guarantee a stable vacuum source: at full throttle there will probably not be sufficient suction to operate the gyros (particularly at altitude).
Pneumatic venturi tubes can be used to supply vacuum for aircraft in flight.
These are bolted on to the airframe and use the slipstream air moving through a venturi to create the necessary vacuum. In extremely slow flight these may also fail to provide adequate vacuum. 
Electric standby vacuum pumps are a popular option as they offer you the ability to keep using your main vacuum instruments regardless of throttle position or airspeed. The primary drawbacks are weight, space (for the installation), and power to run the motor.
Electric gyro instruments are a popular option (though somewhat expensive): They provide a backup for the air-driven system using an independent power source.