Could fast vibrations cause us to travel forward in time
There's two kinds of vibration that would make this "work": thermal vibration and actual shaking. Actual shaking is out of the question because that would involve pushing and pulling a person back and forth such that their average velocity was near-light speed. The "back and forth" part of this would involve way more acceleration, hence force, than the human body could handle.
To have a look at thermal vibration, i.e. heating, we use the following formula for the average velocity of molecules in a body of temperature $T$ (see below for note):
$$v = \sqrt{\frac {3kT}{M}}$$
and in so doing make the assumptions that the body is made up of molecules with the same mass $M$ with uniform temperature $T$. Let's take a human made entirely of carbon-12 for which $M$ is 12 kilograms divided by Avogadro's number. For $v \approx c$,
$$T \approx \frac {Mc^2}{3k} \approx \frac {2*3^2}{3*1.38}*10^{-23+2*8-(-23)} \approx 4.4*10^{16}K$$
which is pretty hot.
So I'm not really answering your question.
Do vibrations work the same as normal movement with regards to time dilation?
Yes.
So a person could walk into such a machine, and walk out hundreds of years in the future, even though a much smaller amount of time would have passed from their perspective?
Well you could do it in principle, and the particles you started with would have travelled into the future, but it would be a stretch to say that the thing you end up with in the future is the person you started with in either method of vibration.
Note: This formula only holds for ideal gases, which a relativistically heated gas is not. But it gives an estimate of the ballpark of temperatures we're working in. If someone has the expression for the hyper-relativistic thermal velocity I'd appreciate a comment or edit :)
In principle, yes, it would work. However, there are two huge practical issues that would probably make it much easier to just fly to a distant star and then come back, or go and orbit a black hole for a bit.
The first has already been mentioned: it would be very difficult to apply the vibrations in such a way that the person isn't immediately liquidized. Under normal circumstances a human being can't take more than a few $g$'s of acceleration, even with a G-suit, and this is nowhere near enough to accelerate them to near the speed of light in a fraction of a second, which is what you'd need to do repeatedly in order to do what you're suggesting.
As Rod Vance says in a comment, it could in principle be done with a time-varying gravitational field, which would apply exactly the same acceleration to every part of the body, so there wouldn't be any stress. However, then you'd run into the second issue: the energy it would take.
Accelerating a person up to $0.99c$ means changing their kinetic energy by $$ \Delta E = \frac{mc^2}{\sqrt{1-v^2/c^2}}-mc^2 \approx 4\times 10^{19}\:\mathrm{J}. $$ (calculation). You'd need to do this many times a second (if you did it only once per second the person would be flying almost to the moon and back on every vibration) so you'd need a total power of at least, let's say, $10^{24}\:\mathrm{W}$. The curent total power generated by humans on Earth is around $2\times 10^{12}\:\mathrm{W}$, which is nowhere near enough. The sun puts out around $4\times 10^{26}\:\mathrm{W}$, so I guess it might be possible using a Dyson sphere, but this would take a huge fraction of an advanced space-faring civilisation's power output just to send one person forward in time. It's difficult to imagine how you could dispose of the waste heat this would generate.
In principle you don't need to use all this energy, because you could recover the person's kinetic energy on every stroke, store it, and then turn it back into kinetic energy going the other way. But again it's very difficult to imagine how to do this. Though in a sense, this is what happens when you orbit a heavy object, so maybe orbiting a small black hole is the best way to do this after all.