This is not possible, there is no way around it because it is intentional that it is not possible (for security reasons) that it is possible to fire an executable
What you can do is try to recreate your .exe as a WebExtension or as a ChromeApp, in the case of ChromeApp you can call it the first extension using link
It was previously possible to use NPAPI to load dlls, but NPAPI was removed.
However, note that you can use NPAPI to load DLLs for example:
The manifest.json should look like this:
{
"name": "My extension",
...
"plugins": [
{ "path": "extension_plugin.dll" }
],
...
}
Use should look something like:
<embed type="application/x-my-extension" id="pluginId">
<script>
var plugin = document.getElementById("pluginId");
var result = plugin.myPluginMethod(); // call a method in your plugin
console.log("my plugin returned: " + result);
</script>