Hello,
It is definitely possible to make a request to a controller from Postman.
In your case, the problem seems to be the fact that the URL is incorrect. In order to get the URL for the controller you can use the following method in your module's main class:
$this->context->link->getModuleLink('<module_name>', 'controller_name');
This will generate a link with the following structure: '<base_url>/<lang>/module/<module_name>/<controller_name>' - the lang part might be optional in your case.
Once you get the actual URL, you should be able to make requests to it from Postman.