https://webbtelescope.org/contents/media/images/2022/035/01G...
Is about 4.5k pixels(as published)/0.04 degrees (2.4 arcmin) a side so 360 degrees of that would be an image with 40 million pixel sides unless I fatfingered something in calculator.
First I assume you mean a '360 x 360 degree' image - meaning you don't want a circle, but a sphere. [0]
Second, much of your image would not be useful, obstructed by the Sun, Moon, Earth, and all that direct and reflected sunlight. I suppose JWST, being in L2, follows the Earth's orbit around the Sun. Over a year, does L2 allow a clear view of the entire celestial sphere?
Third, the number of pixels would seem to depend very much on the chosen depth of focus. I don't know what the zoom levels are, but theoretically it could snap a nice, sharp wide-angle shot of the entire unobstructed sky. (I don't think wide angle is really JWST's mission - more high zoom to spots over 13 billion light years away, but I also don't know much about its optics.)
[0] What is the proper mathematical expression instead of the awkward '360x360 degrees'? what do astronomers and airplane pilots use? I can't think of one atm, but there must be one? A spherical coordinate system for direction, analogous to degrees of a circle?
The whole sky has 4pi steradian.