Why do SDR receivers emit I/Q samples? To my understanding hardware receivers for software-defined radio applications basically take the input signal, mix it with the tuning frequency to remove the carrier frequency and then sample the resulting voltage with a sampling rate that is just high enough for the bandwidth of the payload signal. They emit those samples to the demodulation software in the form of I/Q value pairs. I'm assuming they obtain the Q value by taking another sample $1/4$ cycle (with respect to the tuning frequency) later, effectively doubling the sample rate.

Why do they use the I/Q representation?

I can see how I/Q is a nice representation (in hardware) when synthesizing signals because you can for example do frequency or phase modulation just by varying amplitudes, but this reason doesn't seem to apply to the case of SDR receivers.

So, is there anything gained by using I/Q for output instead of I at twice the sampling rate? Or is it just a matter of convention?
The SDR (or any general digital signal processing system) takes the received RF signal, and downconverts it from the carrier frequency to the baseband.

Now, the real bandpass signal from the antenna does not necessarily have a symmetric spectrum around the carrier frequency, but it can be arbitrary. If the downconverter now shifts the spectrum to the center frequency, the corresponding time-domain signal becomes complex. So, the I and Q samples you get from the SDR are the real and imaginary part of the complex baseband signal, that corresponds to your real passband signal around the carrier frequency. 

More details can e.g. be found at the wikipedia site for digital down conversion.  

To answer your question:

I/Q representation does not correspond to different sampling points of the signal. Instead, it corresponds to the real and imaginary part of the digital complex-valued baseband signal. These parts are obtained by separately multiplying the RF signal with a sine and a cosine and sampling both streams after low-pass filtering. 

Sampling with double frequency can yield the same information as I/Q. It would be neccessary to homodyne the signal to $f_s/4$ to get all information that would have been in the baseband IQ signal to be in the passband signal at $f_s/4$ (where $f_s$ is the sampling rate).