Hello @EoinStokes-4900,
Azure App services might be suitable for you
https://portal.azure.com/#create/Microsoft.WebSite
You will get a url to access the application finally. If your requirement is to have a dedicated IP address, you need to publish the app services in a custom domain. I have detailed the procedure in my blog: http://cloudcompute.info/how-to-configure-an-azure-app-from-marketplace-in-a-custom-domain/
Please mark as "Accept the answer" if the above steps helps you. Others with similar issues can also follow the solution as per your suggestion
Regards,
Manu