There is a closed issue around this issue and ClickOnce deployment.
ApplicationDeployment
class is not available - that class is implemented in System.Deployment assembly
that is not part of .NET Core 3.1 or .NET 5.
We will have a way to expose some of the properties that the class enables, i.e. URL parameters and update status. But, that work could not be completed for .NET 5 release. We will enable it in one of .NET 6 previews. Here are the tracking issues: #27 and #53
Use of ApplicationDeployment
class in your deployed ClickOnce application is not possible, but main ClickOnce experience is available. It includes: creating ClickOnce deployment (manifests), publishing the application, creating and publishing updates. Your application will get automatic update check (and update, if user accepts it) - this part is handled by ClickOnce runtime and the new Launcher tool that becomes the entry point of ClickOnce deployment for .NET Core 3.1 and .NET 5 applications.
Consequently, as for now there does not seem to be a way to do this in .NET Core directly.