It is possible, you would have the choice of a classic on-premises hybrid connectivity with Office 365, or a pure cloud deployment with Office 365.
You use Exchange to have more control over your environment for rules and etc. So you can use a VM on azure but It’s also true that the economics of running an application like Exchange on a cloud platform that charges through resource consumption is unproven for clients.
Office 365 has been successful because it is able to charge a simple monthly fee per mailbox that varies according to available features or license.
But Azure charges for resources like CPU, network, and disk consumption plus the virtual machines that you want to use. It’s interesting to have a look at the Azure online pricing calculator to try and figure out how much some production Exchange servers might cost for your client.