Hi,
Do you mean real apiendpoints using the IPS REST API or regular own controllers inside app/modules/front/... ?
If the latter, then you can use the Application::allowOfflineAccess method to control if the specific page should be accessible even in offline mode.
/**
* Can view page even when site is offline
*
* @param \IPS\Application\Module $module The module
* @param string $controller The controller
* @param string|NULL $do To "do" parameter
* @return bool
*/
public function allowOfflineAccess( \IPS\Application\Module $module, $controller, $do )