With on-off keying, you have a single carrier wave at a constant frequency transmitting its presence or absence. Now I understand that the crystal will not be perfect so you will have some minimal shift over time in the carrier frequency, but assuming the crystal was perfect you would have zero bandwidth because the frequency doesn't change. But datasheets say that the bandwidth is linked to the data rate somehow, what is that equation and what is using up all that extra spectrum if you don't change the frequency?
On-Off Keying (OOK) is equivalent to AM modulation of the carrier with the data bit-stream with a 100 % modulation index.
As a result the entire spectrum of the bit stream is up-converted to the carrier frequency of the OOK modulator. Additionally, due to AM modulation an image of the data-stream exists below the carrier frequency in addition to the translated spectrum above the carrier.
Only an ideal sine wave with constant amplitude and single frequency that runs for infinitely long in time has extremely narrow bandwidth of a single peak. Having any changes to the ideal sine wave such as changing the amplitude means that since it cannot be represented as a single sine wave any more but a sum of many sine waves of different amplitudes and different frequencies, in frequency domain it means it has a range of frequencies. The faster the changes, the wider the bandwidth.
The bandwidth is not due to carrier frequency drift. As a first approximation, the bandwidth is twice the frequency that modulates the carrier:
This may not be intuitive, but consider that the amplitude modulation (including OOK) is essentially a multiplication of carrier and modulation signals, and basic trigonometry tells us that
$$2 sinA sinB = cos(A - B) - cos(A + B)$$
As sstobbe has pointed out OOK is simply digital AM transmission.
Which mathematically speaking means the Signal is multiplied by the Carrier Frequency to give the Modulated Wave (in the Time Domain).
Convolution theorem states that two signals multiplied together in the time domain is the same as the two signals convolved together in the Frequency Domain.
You can model the Digital signal as a Rectangle function.
Long story short that means that you end up with a shifted form of the below frequency graph, with the center around the carrier frequency.