I've found several articles in the trade press along these lines: Microsoft reveals its Internet Explorer successor will support extensions. That particular article goes back to January 2015, and Windows 10 is commercially available, and the new browser is in there... but I still can't find anything more specific than "there will be extensions someday."
I realize that HTML5 support goes a long way, and I understand the power of Javascript, but there are applications that simply require custom CLR code on the client. With no support for ActiveX, and only vague assurances of possible support someday for something that allows "extensions," my question is: Is there any way for the new browser to let me load and call methods in a .NET DLL on the client?
No.
There have been countless security issues with having external plugins, and as a result all major browsers have removed them. I believe you are confusing extensions with these plugins. Extensions refers to things like Chrome's extensions, which are javascript powered and do things like change stlyes on pages, block ads, and so forth.
The closest thing you would get to that would be something like cross compiled C via asm/wasm.