I might be a little bit late to the party however I imagine they will have a business model like Mojo121 has mentioned with some added features. It only mentions new cars in this article however in time I bet they would try the used car market. I think they'd try and mimic the success of companies like Carvana, Shift, Vroom from the US. Selling used cars from an online platform and deliver them with no physical showroom needed.
Carvana alone has an enterprise value of $1.8 billion. I once messaged the CEO of Carvana named Ernie Garcia and asked if he had any plans on expanding internationally. He replied to me and mentioned that they had too much work in the US to complete before they would consider expanding intentionally.
If the Amazon exects have caught on to the success of companies in the US that are using this new type of business model it wouldn't surprise me if they'd like to try dabbling in it themselves.
I do wonder how their business model would differ. For example I'd be massively surprised if Amazon ever started prepping cars to sell, I just can't see that. The dealers would probably stock all of the vehicles and prep them to the standards that Amazon set.
All of that starts to sound a little bit hairy though, if they do pilot in the UK it will be very interesting to see their business model. Will be even more interesting to see how they work with the UK distance selling laws.