When I Google James Webb Space Telescope I get a bunch of videos listed as "Live"
Typically there's a live chat where a lot of stupid people are asking stupid questions.
Their live feed is from the Goddard Space Flight Center.
If you can do without the distracting chat and want better graphics that are more legible go to the whereiswebb.html site at this link on the next line.
[jwst.nasa.gov]
The apparently most up to date information I've gotten from NASA is from their 'blog' page. [blogs.nasa.gov]
It's a different page on the same overall site.
It's of course not up to date but I also enjoyed this 'Inside the James Webb Space Telescope's control room' [phys.org] from Phys.Org.