What would happen if a player fielded the ball with his cap?

Sure, as long as he keeps it in his hand while fielding the ball.

Rule 7.05 states:

Each runner including the batter-runner may, without liability to be put out, advance-

(b) Three bases, if a fielder deliberately touches a fair ball with his cap, mask or any part of his uniform detached from its proper place on his person

(d) Two bases, if a fielder deliberately touches a thrown ball with his cap, mask or any part of his uniform detached from its proper place on his person

In other Words, if a fielder intentionally touches a batted ball with his cap while it is in his hand, all runners, including the batter, advance three bases (ie, every runner on base scores) from where they were when the infraction occurs.