In general, employers do not have to allow union solicitation materials to be posted on company bulletin boards. However, companies also do not have a blanket right to ban all such materials. Companies must be careful about what they ban and when they ban it.
Companies are free to have whatever policies they like about the posting of materials by workers, unions, or anyone else. However, those policies have to be at least somewhat content-neutral. That is, the company cannot enact bans that are clearly meant to prevent union organizing.
For example, imagine two companies, Firm A and Firm B. Firm A has a long-standing policy that says that only company supervisors may post things on bulletin boards and that the things that are posted must be related to the operations of the workplace. Firm B has had no policy and has allowed things like want ads or church fundraiser announcements to be posted. Now a union comes in and wants to post materials on both companies’ bulletin boards. Firm A may clearly deny this request. It has a clear policy that is not meant to discriminate against unions. Firm B may probably not deny the request. It has no clear policy and has been allowing all sorts of material to be posted.
Thus, there is no affirmative right for unions to post their material on all company bulletin boards. However, companies are not allowed to create discriminatory policies that are meant to prevent unions (and only unions) from using their bulletin boards.