@nsa - thanks for that answer. Currently I do all AJAX calls to my backend VB.Net methods - I do not rely on any ASP.Net functionality at all.

Here's an AJAX POST and the backend VB.Net code that receives it. Is that VB code what you have sitting in a container? How do you get your DB online and work with it? I do stuff like make PDF's (with pdfSharp) and create .XLSX files (with .Net XMLSDK).
It doesnt matter whether you use ASP.Net functionality or not, What form is your back end VB code ? are they SOAP web services or REST like WebAPI ? as yes thats the part that would sit in containers.

There are a number of ways you can get your DB online, you can run actually run your DB in a container but also you can run it as a DB service in the cloud. You can work with it using sql management studio or mysql workbench as normal you just need the right connection info.

It a slightly different way of developing as you actually add Docker into your web service VS project so when you run it, it runs inside the docker container rather than IIS, if for instance you used .net core you can even run it on a Linux server

Using containers would require some work to back fit into your current project i would imagine, if you were even considering it i would in the first instance look at finding a tutorial in Dockerizing a Web Service and play around with it

You can just do it more simply and host your web as a web application and your DB as a Database service but form what i have seen that costs more. The main driver of using containers and lamdas and things like that in hosted environments is cost reduction as if your not careful the costs can spiral.

Shoving a VM in the cloud and having it running constantly for instance is really expensive.