From my experience, they spawn about as frequently as Fairhaven's police cars. The Most Wanted racer will randomly spawn in the open world and you will have another opportunity to take them down.
If this happens, continue exploring/free roaming the in the open world. If you aren't able to take down the Most Wanted driver after the race they may manage to escape and disappear from your minimap. Take-downs work the same way as you would destroy a police vehicle or opposing racer in the game - slam into their vehicle hard enough to destroy it. After winning against a 'Most Wanted' driver in Need for Speed: Most Wanted (2012) they will drive away and you must 'take down' their vehicle to receive it as an award.
