You might be able to achieve this if you are using an App Service Environment and an internal load balancer
https://learn.microsoft.com/en-us/azure/app-service/environment/app-service-environment-with-internal-load-balancer
However, based on your description of your current setup I don't believe there is an ideal solution for this as a standard load balancer only allows for the backend ports to map to VMs. Using an Application Gateway might be another option as well
https://learn.microsoft.com/en-us/azure/application-gateway/
I would suggest you make use of the deployment and production slots available that comes a Web App. Once you have the webapp running in the dev slots, test the site to ensure all works as expected. Once it does, switch it to the production slot and reroute all traffic from the VMs to the App Service.
All in all, running an app on a Web App is quite simple. Microsoft takes away the need to manage the VM settings so you can simply deploy and run. I don't see you having any issues simply migrating. The likelihood for issues is small. You can also minimalism it by performing the migration during off hours in case you need to make any changes.
There is also some Web App migration guidance you might find useful
https://learn.microsoft.com/en-us/dotnet/azure/dotnet-howto-choose-migration?view=azure-dotnet