Yes, there are running backs in flag football. They are most often used as an extra wide receiver due to the difficulty of running the ball in flag football. Most plays in flag football are passing plays due to the limitations of the offensive and defensive lines.
Running backs are rarely used to actually run the ball down the field like they would in the NFL or CFL. They are usually used in the passing game as 'checkdown' options, wherein if there is no one open for a pass downfield, the quarterback can pass it to the running back who can try to make a play.
In flag football running up the middle is usually pretty ineffective. Knowing this, most teams run what is called a pitch play. A pitch play has the quarterback toss the running back the ball instead of handing it off. This allows the running back to get to the outside where there is more room to gain yardage.
Every flag football team has a running back. If you want to be one, all you need to do is join a team and try out for the position. Whether you're a kid or an adult trying to play the game there are leagues for all ages.
In regular football to be a running back you need to be fast and elusive, that's no different in flag football. Actually it is more important for a flag football running back to be able to elude would be tacklers. It is far easier to grab someone's flag to down them than to tackle them.