There are a few publishers that allow author self-archiving of book chapters. There is a maintained list of what publishers allow in terms of open access to chapters and books. Some permissions and restrictions are below:
Cambridge University Press
Cambridge University Press do allow archving of book chapters but make distinctions between personal webpages, departmental or institutional repository, non commerial subject repository, and commercial repository or social medai sites. Generally, according to their Green Archiving Policy for Books page the submitted manuscript under review is allowed to be posted to all of these places, but the rules change once it is accepted for publication. One chapter of the manuscript as accepted for publication can be deposited in the author's institutional repository six months after publication, and each contributor to an edited collection can post their single chapter six months after publication.
Emerald permits authors to deposit their chapter in their institutional repository 'immediately post official publication', according to their 'Author rights' page
Oxford University Press
The rules relating to Oxford University Press titles are more complex and described on their Author Reuse and Self Archiving page. In summary, the pre-copyedited version of a single chapter of a book can be deposited to an institutional repository as long as a link to the published version is added once it is available. After publication authors can upload one chapter or up to 10% of a single authored or co-authored book, and a single chapter from an edited book or collection after 12 months for scientific and medical titles and 24 months for 'academic' titles. Contact OUP to establish what their classification of an 'academic' title entails.
Palgrave MacMillan's Policy for archiving in Institutional or Funding body Repositories (Green Open Access) allows authors to deposit their pre copy-edited version of a single chapter of their single authored book, or one of their chapters included in a contributed volume. There is a 36 month embargo.
Routledge/Taylor & Francis
Routledge/Taylor & Francis have released Policies for Open Access Book Chapters which states that chapters from all their books are eligible for green open access. This means each individual author or contributor can also choose to upload one chapter from the ‘Accepted Manuscript’ (AM). Authors are permitted to upload the AM chapter to a personal or departmental website immediately after publication of the book - this includes posting to Facebook, Google groups, and LinkedIn, and linking from Twitter. However there is an embargo period of 18 months for HASS and 12 months for STEM book chapters if posting to an institutional or subject repository or to academic social networks like Mendeley, ResearchGate, or Academia.edu.