I have a programming contractor - how do I record their screen remotely?

A lot of remote access software can do screen recordings, I don't know how they would be designed for saving the file, they would either be local (to his machine) or if its more a central management type there might be a server it could be sent to.

Since you wanted to him to initiate it, I'm assuming this is because of somewhat flex hours of work incase there is no one at the office to start it from your end.

You might need to do something like having a computer in the office that he remote logs into first, console session login, then establishes a remote session to his work station using something like GotoMeeting or GotoMyPC, hits record. Then he logs off the workstation in your office and just works away for the day, if you want to see what's up just check out the local workstaion. Because it was initiated from the office the file when the recording happens should be stored on the local PC.

It has a 30 day trial so you can see if that setup would work for you.

Another idea if you didn't need "real time" would be ad hoc, where you put in remote access software on his workstation and you can login anytime you want to see what's happening and record using any local screen recording software needed. This might be easier since LogMeIn free would work for this. Then you can do spot checks say a couple of times a week to keep the investors happy.

I would recommend that you learn to trust your developers and judge them on the amount of good code they produce, not whether they stopped working to play a few rounds of Solataire while mulling over a problem.

Anybody who would work in an environment where they are so untrusted that screen captures of their screens are stored on a server for other people to spy on them is either desperate for work or secretly knows that they can't be trusted. Either way, they won't be any good at their jobs, or they'd go someplace where they were treated like human beings.