Do familiars advance per season like companions and magi? Do they get 7/4/2/2 like grogs? Or is there a mechanic I am unaware of?
Per-season, typically.
Generally a familiar is handled mostly by the magus whose familiar they are, but will often get played by someone else during the game.
You can also advance some grogs per season. There is no need to always apply the simple generic method proposed on Grogs p.67.
If you care to do this extra work, just make sure that every season enough grogs do what needs to be done for the covenant.
Cheers
As One Shot indicated, either work.
Just remember that grogs (like most companions) work for 2 seasons a year, and so only recieve Exposure for those seasons.
Covenants that spend the resources to eg train their grogs in the remaining time, will have more competent grogs.