You can require "no moonlighting" as a term of employment so that an employees focus is on one job alone.
You can with limits. But you have to have a set reason. Generally companies simply use the "three Ps" concept. Employees have to be present, prompt, and prepared. Or they use non-compete clauses. If you simply say that someone can't have another job, you would have to give a negative result from their other job if you let them go. Otherwise, you are infringing on their right to seek opportunity.
Baseball is obviously different than the NFL in that it is a 162 game season with games throughout the week and the season runs for most of the year with spring ball and post season and other events and seasons. MLB umpires may also work games like the Dominican leagues. MLB umpires are full time because they have actual full time jobs. Still - several MLB umpires DO have other jobs or companies that they work in the off season.
It is kind of like a teacher. Most teachers wouldn't have other jobs during the school year because of their 5 day work week not because they aren't allowed to have them. Sunday school teachers on the other hand, have other gigs during the week.
That's not to say that I wouldn't want the NFL to model the higher training standards set by baseball. Unfortunately, the constant training, multiple semi-pro leagues, and 162 game season, make it pretty hard to compare to football.
But the stuff in the post by
@Akrasian above seems like a pretty logical thing to do. If they can't keep their other job in doing those things, oh well. Commit to one or the other.