I am trying to link the Azure Functions deployment to integrate with Visual Studio Team Service however once I select VSTS as the source, it doesn't show any other projects.
I have other web apps in my account which feed from a VSTS project.
The reason this is important to us is because of the workflow. We currently use AWS API Gateway + Lambda for our webhooks however it doesn't fit our development/deployment workflow.
When you select "Visual Studio Team Services" as deployment source from New Azure Portal, it is using Kudu service which is designed for Git version control. So it will only show the projects with "Git" repositories.
Kudu is the engine behind git deployments in Azure Web Sites. It can also run outside of Azure.
If you want to deploy the TFVC project, try with Azure Classic Portal. You can refer to this article for details: Continuous delivery to Azure using Visual Studio Team Services