Hubble is one of the few telescopes that could possibly image Starman, as a dot. James Webb will have far more sensitivity. It's a really small space rock in a really big space. You can take a look at https://www.whereisroadster.com/ which has information on how big of a telescope it would take to see it, and to resolve it, updated in realtime. The answer right now is:
A telescope about 7,223 ft (2,202 m) in diameter would be required to resolve the Upper stage from Earth. A smaller one could see him as an unresolved dot, about 13.9 ft (4.2 m) in diameter, in ideal conditions.
Hubble, being outside of the atmosphere, is better than the small unresolved dot scenario. There are telescopes right now on the ground that if they pointed at it and spent an hour of imaging time could probably find it, but those telescopes aren't likely to be used for this purpose, their primary purpose being scientific pursuit.
To get an idea of what something might look at, I suggest looking at the Hubble pictures of 2014 MU69. Starman is of a comparible side, but would be moving faster than 2014 MU69.