I know an individual can use the natural frequency of glass to break it, but is it possible to break glass with a frequency other than the natural frequency? Will the glass still vibrate if the frequency around it is not its natural frequency?
Reductio ad absurdum - place the glass in a beam of sound of sufficient intensity and it will shatter if it does not melt first. An ultrasonic siren is a good place to start.
An object like a glass has multiple harmonics. Exciting any of these will cause resonance. During resonance you can build up the amplitude of response more than when you use an off-resonance frequency.
It is actually quite hard to real a glass by exciting its fundamental frequency - and even harder to do so with harmonics. The problem with the higher harmonics is that the side from which the sound arrives will have multiple nodes/antinodes, so your pressure wave will have a hard time exciting a pure wave (it would be pushing in the right direction in one place and the wrong direction in another- because the phase requires from the exciting wave would change with position).
The following diagram might illustrate what I mean:
On the left is a glass resonating at the fundamental frequency: on the right, it is oscillating at a higher harmonic. You can see the green state (antinode towards the sound source) and its opposite, the blue state (antinode away from the sound). But you can see that a short distance along the wave front, the phase needed to excite the resonance is "wrong" - so it will be hard to excite these higher resonances with sound. You can, however, excite them with a more local force (e.g. by gently tapping the glass)