For small sections with high wear, vinyl won't be very cost effective or durable. Porcelain tile would be a better material. Either way, the floor needs to be flat and smooth.

For your application, the LVT type of tile, like the Allure would be the easiest to DIY as it sticks to itself as a loose lay. However, it will need some type of anchoring at the edges to stay stable since it's such a small amount. Transition strips screwed into the subfloor over the vinyl would be your best bet for that.

If you decide to do tile, then what needs to happen depends on your subfloor type and how stiff it is.

Since most sheet vinyl flooring is sold in a 12 foot width, there would be a cost issue. However, sheet vinyl of a high quality can be a very durable and attractive flooring option that has the clear advantage of being seamless.

Big box stores often sell smaller remnants of sheet vinyl that are cheaper and easier to work with.

Katie: the basic process is:
1. Remove carpet: Cut out the carpet with a utility knife. It will have padding underneath it, which needs to be removed as well. There will be tack strips along the wall that you'll need to take out.
2. Prep the subfloor: Are you on a slab or basement/crawl space with a wood subfloor? Details on doing this vary depending on your subfloor.
3. Lay the vinyl: Cut the vinyl to the precise shape, using regular scissors. There are inexpensive vinyls that you just tape down, using a special double-sided tape, around the edges.
4. Restretch the carpet: You need tack strips along the new edges of the carpet, and the carpet attached to them.

Hope that helps!