Employers are required to post certain information. They do not have to allow workers to post their own bulletin boards.
First of all, the federal government of the United States requires employers to post certain information for employees to see, though not necessarily on a bulletin board per se.
[F]ederal employment laws require employers to post notices ... that are conspicuous and accessible to all employees. (shrm.org)
This information includes laws, and especially health and safety information. For example, all restaurant employees must wash their hands. Employees handling hazardous waste need proper protective gear.
There are also often state or local government requirements, such as local minimum wage laws. The purpose of the posting requirement is to make sure that employees know their rights. If the employer does not post these rights, they are in violation of federal law and can be cited by OSHA. States also have their local enforcement agencies.
OSHA citation may be issued ... w/ violations are listed. (eNotes)
The importance of having a bulletin board might seem minor. However, employers who try to take advantage of workers who do not know their rights are playing a dirty game. It is important to workers’ safety and well-being. For example, if workers do not know it is their right to take a lunch after working a certain amount of hours, employers can force more time out of them and the employee’s health may suffer.
Employees can certainly ask employers to put up a bulletin board for other announcements. Contacting the OSHA representative is a good start.