You can host any app that runs on IIS in Azure directly as an Azure App Service. An app service is basically a hosted web server. Out of the box it supports most frameworks including .NET, NodeJS, python and Java. Basically you create the app service, configure the server to use the version of the platform you want and then deploy your app to the root of the site. WordPress sites would work here as well. For sites that require a SQL Server database then you can also use Azure's SQL instances.
If you need to run a platform that isn't supported in Azure App Services then you can upgrade to using an Azure VM. With this approach you are completely responsible for the VM so you can install whatever software you need in whatever format you want. If you need a DB that isn't supported in Azure then you can also set up a VM to host the database as well.
In general the more SaaS you use the less you have to manage but the more limited it becomes. You mentioned WordPress so those sites could be hosted in Azure App Service (google for some discussions). WordPress uses MySQL which is now supported on Azure so you could use Azure for MySQL to host the DB. The biggest concern is cost. If you have many WordPress sites then it might be cheaper to host them on a dedicated WordPress provider rather than in Azure. You'll have to decide that for yourself.