Partial, and a bit simplistic, I compared it to Juno, currently orbiting Jupiter.
Like Juno, JWST is not orbiting Earth but rather the Sun (and or L2 itself).
So, like Juno, a TLE (list of orbital elements of an Earth-orbiting object for a given point in time) cannot exist for JWST (aside for the brief moment it was considered in Earth orbit, then the USSF assigned a TLE) as it does not orbit Earth.
JWST TLE briefly assigned:
https://twitter.com/planet4589/status/1475148763954094083
SpaceForce TLEs for Webb showing it in a 316 x 1011924 km x 4.1 deg orbit. No TLE for the rocket stage so far.
TLEs for the ZY1-02E launch showing four objects, three in 761 x 766 km x 98.6 deg orbit and one (rocket stage) in 598 x 764 km
4:57 PM · Dec 26, 2021
Currently, US Air Force Space Command still has a TLE for JWST though..
https://www.n2yo.com/satellite/?s=50463
Consider Juno, no TLE:
https://www.n2yo.com/satellite/?s=37773
So, in answer to
Does the James Webb Telescope have a Two Line Element set?
Yes, but it shouldn't ..?
Also see:
Has a TLE ever been issued for a spacecraft trajectory not bound to Earth orbit?