What is there now? What is at the bottom of the stairs? What flooring is in the rooms off the upstairs hallway?

why are you concerned about matching the carpet? it can be anything you like.

I’m a bit mixed up - will the second floor have some carpeting and where does it end and vinyl begin? If you don’t have a choice, be sure the stairs and rug relate to the vinyl.

When you say vinyl tiles I am picturing the vinyl tiles they put in kitchens 40 years ago. I assume you're talking about LVT however. Do you have a sample of what you're considering so we can get a better idea of how this would work? Like jck910 said, what's on the floors in the bedrooms off the hallway? We need more context.

I will bet the treads are just plywood under that carpet so be preapred for new treads before runner . As for the vinyl tiling I have no idea what that is exactly. IMO stair treads are part of the lower floor so what ever you have on the floor at the bottom of the stairs is the foor you worry about matching. BUt check first to see what the treads are under the carpet.

Downstairs hallway has this floor throughout.

Having trouble posting on here. I've just managed to post the photos above and given a little more info in my original post.

The treads are wood not plywood, not MDF.