Can ASP.NET MVC websites have middleware?

Yes, although it seems the documentation is intimately linked with adding ASP.NET Identity to your site.

Adding ASP.NET Identity to an Empty or Existing Web Forms Project - ASP.NET 4.x | Microsoft Docs[^]
Adding minimal OWIN Identity Authentication to an Existing ASP.NET MVC Application - Rick Strahl's Web Log[^]

The key part you're looking for is OWIN. You'll need to add the Microsoft.Owin.Host.SystemWeb package, and add a Startup class to configure your OWIN pipeline.

using System;
using Microsoft.Owin;
using Owin;

[assembly: OwinStartup(typeof(YourNamespace.Startup))]

namespace YourNamespace
{
    public class Startup
    {
        public void Configuration(IAppBuilder app)
        {
            ...
        }
    }
}