According to the NASA blog on Dec 26:
Separately, overnight, the temperature sensors and strain gauges on the telescope were activated for the first time. Temperature and strain data are now available to engineers monitoring Webb’s thermal and structural systems.
Presumably the temperature data is currently (as of Dec 28) being calibrated as the sunshield unfurls, and will be streamed to their website when it is normalized.
edit: The temperature is now showing, accompanied by this message:
There are many more temperature sensors on the observatory that our engineers monitor throughout the deployment, commissioning and operations processes. The data points shown here give a good overall indication of the temperature trends on each side of the sunshield as we move through deployment and commissioning. They illustrate the great contrast between the hot and cold sides of the spacecraft and the incredible engineering and effectiveness of the sunshield. These temperature observations are reported daily from actual spacecraft telemetry data.
I think this supports my guess that there are a bunch of sensors and it took a few days to turn them into data they found acceptable for the website.