We have an Active Directory domain sitting on a networked Azure VM and have a separate Azure WebApp/Website that is running Orchard CMS on the same virtual network.
Do webapps support windows authentication for connecting to Active Directory? If so, how do I go about setting it up in IIS?
No, it's not possible. In order to have Windows Authentication is something for on-premise deployments. For Azure Web Sites Azure Active Directory is clearly the best option. Sync from AD to Azure Active Directory is also quite easy to setup.
If you still want to absolutely use Windows Auth and host your website on Azure, you can create Windows VM and host your website there. You then need to join the VM to your AD. To this, both VMs must be in the same network. So if your VM is on-premise you will need to create an site-to-site VPN