You might have some luck checking out your nearest flooring store and seeing if they have end of roll or remnant piece of carpet. You may be able to find a piece that will fit and have it cut and bound to serve as a little mat for that vestibule.

There are stores that can order in more heavy duty pieces of carpet that are made for entryways & vestibules as well.

Good Luck!

I hadn't thought of binding it, thank you, great idea!

I would install some vinyl sheet flooring and add a door mat that is washable