It's not that close. The moon is 384,400km away, and JWST will be 1.5 million km away (i.e., around 4x as far as the moon). It would be a pretty serious undertaking to get to it and get back.
That being said, and I haven't done the math, but I think a Falcon Heavy and Dragon could do it at least in terms of delta V; although there are numerous problems with that like the fact that the dragon doesn't have a proper airlock for starters.