Would it be an ASP.NET web developer or would it need to be someone
specifically trained in Umbraco?
Ideally both.
Since you talk about the need for maintenance and development, the primary requirement is knowledge of C# and the .NET framework, along with a number of basic web technologies: HTML, CSS, Javascript, etc.
If it's a very old (version 4 and below) Umbraco, knowing XSLT is good. Experience with MVC is great for version 7 and above.
Knowing about Umbraco is actually required in order to get any work done - but it's not that difficult to pick up (especially not if you already have experience with CMS development).
Let me put it this way: It takes longer to teach an editor to be a programmer, than it takes to teach a programmer to be an editor.