What are usable entropy sources in a headless machine without /dev/random or any cryptography API? If I have a headless machine (no mouse movements, no keyboard presses, or user input), without any cryptography CSPRNG APIs, where can I collect entropy from?
I believe that on any CPU that you should be able to find at least two sources of noise with physically proven behaviors to create a purely random number.  The first is through oscillator sampling, assuming that you have a fast clock and a slow clock.  The second assumes that you have DRAM, and through the two-way shot noise in the channel, you should be able extract random bit flips in DRAM if you can control the timing.

Clock Sampling

The clock in a system is not a precise as you'd hope.  Generally, you have a core clock that is generated through a PLL, and you also have some sort of realtime clock at 32.768kHz because that will overflow after bit-15 to give you a 1-second pulse.   By sampling the fast clock with the slow clock, you can generate a series of bits; however, there's a few caveats:


oscillator jitter alone is not enough to create randomness
you can actually degenerate the randomness by trying to make it random if you have a poor circuit model.


As an x86 was mentioned (I know the least about this CPU btw), the time stamp counter combined with a 8254 on older machines or HPET on newer machines will get you a fast and a slow clock source.   Michael S. McCorquodale dissertation should have a very good analysis on the clock jitter in both cases, or at least he did at this defense.

DRAM

If you can control the timing of the DRAM, you very likely could get a random noise source due to the fact that the noise in a semiconductor channel is a function two-way shot noise (and not classical Johnson Noise). Let's limit the scope to a single bit.  You would have to put charge on the DRAM capacitor and then gradually delay the timing until that bit gave you a 1 or 0.  The charge amplifier on the DRAM rows would also add to the noise, but I will ignore that component for now.  Carver Mead and Rahul Sarpeshkar both use shot-noise in their neuromorphic engineering papers as a source for entropy on silicon neurons, and circuits people can derive it but I really don't know where an external reference is.  The caveats:


DRAM timing control 
assume that you are not near "freeze out" as the shot noise goes away.


I believe that either of these, combined with a hash, could be used as a means to generate a true random number as the noise sources are rooted in physics.  You'll need to grab a cryptographer to take what I described to the next step.