There's no obvious or hidden rating as such. From observation, a player is more likely to foul if his defensive ratings are low, obviously for instance if he has poor tackling. Number of fouls also depends on position played.
Fouls themselves are not, IMO, a big deal, other than the fact that when you foul the opposition retains possession of the ball. Even yellow cards are not that big a problem - TO ME - as long as a player does not tend to pile up too many of them, since two in one game equals a red.
I try to stay away from players who red card, no matter how skilled they may be. Going a man down is just too much to handle unless your team is much stronger than the opposition.
And one thing we do know is that players' tendency to yellow and red is governed in part by a hidden rating. This was discussed in a long-lost developer chat somewhere, I am sure of it.
stews, in your particular case, being a startup team, you shouldn't really worry about your fouls. Observe and learn the game, build your stadium and pile up cash, don't sign any players long term and work toward acquiring better players in your second season.