Not really, at least through the Sonos software. Apple has blocked apps that use the iOS device as a hard drive (like Sonos does, and I think others). You can, however, use AirPlay 2 to ‘cast’ the music, using another app on the Apple device, to certain AirPlay 2 capable speakers. 

But your question doesn’t scan, at least the way I’m interpreting it. Sonos Radio HD doesn’t exist on your iPhone. The data exists elsewhere, and is streamed by the speakers, and the remote control that is running on your phone has little to nothing to do with that stream. 
 

Hi Bruce,

I appreciate your response, but I’m more curious about the Sonos HD service. Does it work like Spotify or Pandora? Reading between the lines of what you’ve written it sounds like the service is a subscription for your speakers, not for you as an individual.
So could I continue listening to a Sonos radio station in my car? From what I understand the answer is no.

Sonos Radio and Sonos Radio HD only plays directly on the Sonos speakers.  You can’t play Sonos radio directly on an iPhone or android device.

The only way you could listen to Sonos radio in your car is if you setup a WiFi network in you car and strapped a move in the back seat.

Would the Sonos Move needs a car seat? Haha. I wouldn’t want to break any state laws. :) 

On a serious note how does the Sonos streaming service work for the new headphones? Knowing these are connected via Bluetooth. 

Strapping the Move in is recommended, though not a legal requirement.  It’s not that I am worried about damaging my speaker, I just don’t want it flying at my head in case of accident.

Sonos Radio doesn’t work with Ace.

Well, you could technically play Sonos radio on a Port, without output over RCAs to a TV, connected to an Arc, that linked to an Ace.  That would work.  I imagine that’s not ideal for most users.