I understand that JWST will have a vertical elliptical orbit around L2, but what I don't understand is how the telescope will actually maintain an orbit if there is no body in L2 to actually orbit around. There is no center of mass to pull the telescope inward to maintain a circular orbit around L2 while also orbiting the sun. Does the telescope maintain this orbit using its own power?
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$\begingroup$ L2 is an equilibrium force point. Unstable, but an equilibrium point. $\endgroup$– AtmosphericPrisonEscapeCommented Jan 10, 2019 at 21:11
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Basically, it would not be orbiting L2 in a Keplerian orbit sense. There is no mass at L2 for the spacecraft to orbit, you are correct in that sense.
The JWST would be in a solar orbit in that would normally be a little longer than one year and would use the contours of the gravitational fields from the Earth and the Sun around the Earth-Sun L2 point to keep it moving around the sun in a predictable region near the L2 point, and that deliberately planned gravitational perturbation is what will keep the JWST in a 1:1 resonance with Earth around the Sun, (along with some very small mounts of station-keeping from its maneuvering thrusters 1, 2).
As a result, when seen from the Earth, the satellite would appear to be moving in a curve around the L2 point.
https://en.wikipedia.org/wiki/James_Webb_Space_Telescope#Orbit
Lower your volume before watching (cued at 05:31):
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$\begingroup$ Great answer! I hope you don't mind, I've added a few more links, feel free to roll back or edit further. $\endgroup$– uhohCommented Jan 11, 2019 at 2:10