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.