Does NetJets have any contracts to do charters with anybody or is it a free charter company?
It’s a fractional (“timeshare”) operation for the most part. See the NetJets web page for more information.
By “free charter,” can I take it you mean anyone can charter their aircraft? They don’t offer charters for free. 
You assumed right.
I think you can charter them and buy one of their low time cards.
By buying a prepaid Marquis Jet Card, is the only way to charter NetJets aircraft if you are not a fractional owner.
Marquis Jet is a fractional owner of NetJets aircraft, so by buying a prepaid card, you are flying as a Marquis fractional owner.
They usually sell 25 hour cards as a minimum, but I’m sure they’d work something out.
R. Federer, the famous tennis player, is in a commerical for them. I don’t know if he has a contract with them, but using him is a powerful marketing tool.
I was asking because one of them seemed to go along with a team’s schedule.
Is there a list of people that have contracts with them?
Could be the team’s owner going along… Or the wife/family of one of the players.