What is the lowest temperature tungsten starts sublimating at?
Well, you see, searching online for "vapor pressure at such-and-such temperature" (or conversely, "temperature to get such-and-such pressure") is often fruitless, and unnecessary too, because pretty decent approximations are known. Here is one:
$$\ln\mathfrak p=-{\Delta H\over R}\left({1\over T}-{1\over T_0}\right)$$
where $\mathfrak p$ is your vapor pressure in atmospheres, $\Delta H$ is the enthalpy of vaporization and $T_0$ is the boiling temperature at normal pressure. For tungsten, the latter two are reported as 774 kJ/mol and 6203 K, respectively. Now you may plug in your desired pressure and solve for temperature.

That's a relatively crude approximation, for the enthalpy will hardly remain constant over such a wide range of temperatures. This page may contain better approximations designed for specific ranges.