I have a 2002 Ford F150 with a broken actuator rod. Can I bypass the actuator rod with push button starter? Thank you.

 

It is not that it cannot be done ,but if the rod is not working the ignition switch is also controlled by that rod.
What would be required would be for an entire ign sw and wiring be installed in a suitable location and that can be just as much trouble as repairing the original problem.
Not knowing the condition of your vehicle you might depending on it's condition consider getting the parts from Ford and correcting the problem. A lot of shops chose to just change the upper strg col because they didn't want to tackle the job.
It is not a difficult job, but requires some special tools like a strg wheel puller and careful removal and installation of the new parts. If you have the time and can tie up the vehicle for the time it would take you or whom ever you get to do the job when done your vehicle would be as it should be. Usually the rack attached to the rod goes south and the rack and gear are replaced.