Regarding the title: "When do we expect blackbody radiation", the answer is everywhere (emissivity notwithstanding).

At our maximum temperatures (tera-kelvin in heavy ion collisions), we get thermal pions radiated (I'm not sure about electromagnetic radiation--the timescales may be too short).

Ultra dense systems such as core-collapse supernovae (100 GK) radiate thermal neutrinos.

From there we have hard and soft X-rays in the various stages of nuclear fireball formation (100 MK - 285 kK)

Then lightning are arc-welding in the tens of kilo-kelvin emit UV.

The Sun is our best known visible-spectrum emitter (5772K). Vis. cuts of at the Draper Point (798K).

Then IR ofc, and from 300K down to 150K is the domain of microwave radiometers for weather and atmospheric profiling.

Down to 2.72548K, the CMB.

In microwave radiometery, emissivity plays a huge role in recovering physical properties of the surface and atmosphere, and in those systems, we speak of "brightness temperature", $T_B$ at a specific frequency ($\nu$), which satisfies:

$$ B_{\nu}(T_B) = \epsilon(\nu)B_{\nu}(T_{\rm physical}) $$

where $B_{\nu}(T)$ is the blackbody spectral radiance.

Spaceborne systems (ATMS, SSMIS, GPM, AMS, ...) report data as $T_B$, and users then apply their models to recover $T_{\rm physical}$ and $\epsilon(\nu)$


Regarding medical thermometers, idk how they are calibrated, but The Web says the emissivity of human skin is $0.98$, which is an $11^{\circ}\,$F correction at $98.6^{\circ}\,$F so it definitely matters.


Edit: since the tag says "atmospheric science", the standard is 

https://www.researchgate.net/publication/299103021_Microwave_Radar_and_Radiometric_Remote_Sensing