I don’t think the news outfits can legally prohibit their employees from contributing to the candidates of their choice; that’s their constitutional right. However, I do believe that they should drop their pretense of being objective news agencies. We should go back to the way newspapers used to be for the first hundred or so years of this country, where they proudly declared their political leanings in their names, such as “The Arkansas Democrat,” The Missouri Independent,” “The Republican Statesman,” etc.
Generally, employment at private companies is "at will" for both the employer and employee. That means the employee can quit at any time and the employer can fire the employee at any time. No reason required.
Now, there are certain areas that are protected. For example, if the employee is covered by a collective bargaining agreement or if the employee alleges various kinds of discrimination based on minority status, "whistleblower" activities, or the like.
There's nothing in there about political activities. Just as Ford can demand that their employees not buy foreign cars, a liberal organization could demand their employees not be conservative (or vice-versa).
After all, though you have a constitutional right to free speech, there's no constitutional obligation on anyone to give you a job.