Cowchick and Beau are both correct on several points. Typically the pick up men are part of the Rodeo Company/Stock Contractors crew. Most that I have known have a minimum of 3 horses for one performance as it is really hard on the horses to try to do it all each night. You need to be a proficient roper, cow hand and horseman. You must be able to react quickly and think on your feet. You must also have the confidence to run into a bad situation while everyone else has the urge to run away. Pick up men don't usually just ride during performances, they care for the stock, unload/load and haul the stock to their next destination (so they have commercial licenses to drive big rigs) and some companies even haul in the arena and set it up. I use to watch one crew pull in before dark on Thursday nights, set up the arena, unload the stock and be done by sun up, sleep for a couple hours get up and work all day to prepare for the show, sound systems, arena grading, meeting with local code enforcements, chambers of commerce, PRCA officials, judges, etc.
It is not easy work and it getting into the job takes a really long time. I use to warm up and cool down horses at rodeos for announcers, contractors and pick up men and the pick up men had to be able to jump on any horse the contractor had available should one of their own become injured or just having a bad night. There are no excuses for them to not preform their duties.
If you choose to do this then be prepared for a lot of hard work at the bottom to pay your dues and I suggest hitting the gym, decent upper body strength will not do. If you have a 230lb saddle bronc rider grabbing onto you while coming off a horse at 20 mph your in for a real treat. You need to keep working on your roping and you need to have a lot of knowledge about rodeo, the rules, procedures, laws, politics, economics, etc of all that rodeo entails. There is a lot going on behind the scenes that most people never know about.
As a woman is it possible? I believe it is. It is easy? Not a chance and never will be, it takes a lot more than a good horse. Good luck to you and I think you should take advice from people like Cowchicks husband.