While not an amendment, there certainly have been provisions to compel entities from providing a forum for sides they don't want to promote. [0] See also the now-repealed fairness doctrine. [1]
[0]: https://en.wikipedia.org/wiki/Equal-time_rule [1]: https://en.wikipedia.org/wiki/FCC_fairness_doctrine
Indeed, it's my position that such laws would in fact be infringing on the free speech of those private companies. In addition they might cost those companies money, making these hypothetical laws also anti-free market.