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As far as I'm aware, traveling at $c$ will prevent time passing due to time dilation. Electromagnetic waves rely upon oscillations to propagate. Since oscillations rely upon the passing of time, how does an electromagnetic wave oscillate if time does not pass?

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Electromagnetic waves rely upon oscillations to propagate

Electromagnetic waves are propagating disturbances (or oscillations) in the electromagnetic field and the electromagnetic field exists everywhere and everywhen.

But the electromagnetic field itself does not 'travel' at $c$ so your reasoning isn't clear to me.

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you are correct that the EM field propagates at c, but as the comment from Acuriousmind mentioned, there is no frame of reference traveling at that speed. What that means in simpler words, is that the equations break or diverge for anything traveling at the speed of light. Light is not bound to time dilation or length contraction for that same reason. Only objects traveling close, but never at, that speed are.

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EM waves are mass-less. The time dilation effect you mention only applies when a travelling body has mass.

Note that for a body that has mass it would take an infinite amount of energy to travel at c.

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