Retrieving menu item using REST
There is still no native way within Drupal core to expose config entities via REST. There are 2 contrib modules which claim to allow retrieving menu items via REST, neither of which I have used.

REST Menu Items

Allows you to specify maximum and minimum depth in the request.

REST Menu Tree

Returns the menu tree. Not yet covered by Drupal's security advisory policy.

Update 28 March 2017

REST Menu Items works as advertised but check the issue queue for any outstanding bugs. For example, currently the alias for a menu item isn't returned.