There doesn't seem to be a client available, or maybe I'm just looking in the wrong namespace. How is this expected to be done, or is the answer that I have to find another message service?
MSMQ is part of Windows/Windows Server and is therefore not platform independent and part of vanilla .NET Core. If you want the System.Messaging framework, you need to target the .NET Framework. It's been a little confusing getting to this point but this is a good example of the inherent differences between the two in the new Microsoft world.
For Microsoft's explanation, please see https://docs.microsoft.com/en-us/dotnet/articles/standard/choosing-core-framework-server