Can I take DC directly from W1209 Thermostat board?
Probably no.

Note that while Q1 is switched off there is 12 V at both sides of the relay coil. That is probably not what you need.

It could be worse than that: at the moment that Q1 is switched off, there will be a bit of a voltage spike which could (unlikely, but I haven't done the calculations) go through your other 12 V DC circuit and destroy it before the diode helps diminish that spike.

How to make it work.

Refer to the STM8S003F3 datasheet.

Once you have consulted the datasheet to find how much current can be sourced from pin 10 of the IC, you can determine if it would be better to use one buffer (to go to the 12 V DC control circuitry) attached to pin 10 of IC1, so that the maximum output current is not exceeded, or if pin 10 can't cope then two buffers where the additional one is put between pin 10 and R7.