Hello Kate,
According to the current REST API documentation it appears that the only related call is for messages (readonly/set). For example
http://community.lithium.com/community-name/restapi/vc/messages/id/1271/readonly/set
If the client know which board they want to set to readonly they should be able to create a script which gets each message from the desired boards and user the "readonly/set" API call with the message id.
Otherwise, they should be able to go into the community structure tab and while editing each board check the "Make all posts read only" box.
Let me know if you have any further questions.
