What is the vapour pressure of carbon at room temperature?

The vapour pressure of carbon at 3642°C is one atmosphere (this is the sublimation temperature given in Wikipedia).

At lower temperatures, the vapour pressure will be less, but of course it will never be exactly zero.

However, I have not been able to find a reference which gives an actual or estimated value for the vapour pressure at (let's say) 20°C. The consensus seems "too low to be interesting", which leaves it lazily open as to whether it is $10^{-6}$ torr or $10^{-20}$ torr or whatever.

Can anyone help, or point to a suitable reference?

You can estimate it using the Clausius-Clapeyron relation, which can be rewritten for the vapor pressure as $$p_v = p_0 \exp\left(-\frac{E_v}{R}\left(\frac{1}{T}-\frac{1}{T_0}\right)\right),$$ where $R=8.314~\mathrm{JK^{-1}mol^{-1}}$ is the universal gas constant, $p_0$ is the vapor pressure at a given temperature $T_0$, and $E_v$ is the heat of vaporization (in J/mol). With carbon, take $p_0=1~\mathrm{bar}$ and $T_0=3915$ K, $E_v=117$ kJ/mol. Then at room temperature (293 K), you'd expect $p_v=$ 5e-20 bar = 5e-17 hPa.

Typically, this expression is not exact due to subtle temperature dependences of the entropy, but it should be reasonably close.