I have a Gun Mettle campaign coin and have currently done 3 contracts. Two right when it came out then a third yesterday. I think I read something stating two contracts will be available per week, but when (i.e. what day(s) of the week) do they refresh?
You'll be assigned two skill-based contracts each week for the length of the Gun Mettle Campaign.
This was copied from the official website of Team Fortress. So yes, 2 new contracts every week, there is no specific day one will be released.