Most of the platform native features, like the one you're suggesting, may be exposed to the Flutter app by using Flutter Method Channels.
Enabling your app to take actions based on voice commands requires the iOS part to implement SiriKit intents handling. You have two options for implementing SiriKit intents handling:
Take into account that, as far as I could get from the documentation, Flutter Method Channels are not available in App Extensions without a UI, since the channel communication mechanism relies on the presence of a view controller. If your not implementing an App Extensions with UI features (using Flutter inside the extension, which may not always be a viable option), the only viable option is to adopt the in-app intent handling strategy.
Handling SiriKit intents in the app requires your app to enable multiple scenes support. This requires some adjustments to the iOS generated code of your app.
I needed my app to respond to the user asking Siri to "Play on " and I've developed a Flutter plugin to do that. You could use the plugin source code as a starting point for implementing the intents domain relevant to your app (bear in mind that you can also define custom intents, if no Siri built-in domain suites your purpose).