HTML5 is still a draft and does not have much difference in syntax from SGML, but it also allows you to use more XHTML style code (for example, using self-closing tags).
Last time I checked, Visual Studio 2008 and 2010 released XHTML-compatible code, i.e. with proper private tags. It does not have canvas and video support, but that is not your question either, is it?
So, if you ask how to change the default template for HTML and ASPX files, I don't know. But if you ask about a designer for HTML, and ASPX generates compatible code, then yes, it is.
Addition: I just created ASP.NET WebApplication in Visual Studio, changed the doctype to <!DOCTYPE html> and confirmed the output to http://validator.w3.org/check using HTML5 validation.
Result: This document has been successfully validated as HTML5!
In short: Yes, VS2008 and VS2010 support HTML5 out of the box; -)
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