No you cant see your missions or contracts during game
But The Bridge allows player to rejoin the game
So you can technically leave a game, check on your stats, missions or contracts and rejoin
One of the way I keep track of my progress during missions/contracts without leaving is using stats kills msgs
Just note the end stats you should be on for completing and play till you reach it
This way you dont have to leave the game
FYI, you need gold rank for stats kill msg