I'd put a floor mat down in any car to protect the carpet, and in a Morgan where it's the passenger's heels that are mainly impacting I'd think it's doubly important.
I'm not a particular fan of carpet in open cars so I bought a length of fine-flute rubber mat from
https://www.carbuilder.com/products/fine-flute-rubber-matting-per-metre, dead easy to cut it to size with a Stanley knife.
If you want to stick with carpet you can get everything you need from Woolies, they do carpet in a range of colours. Again, very easy to cut to size but you'd have to find someone to bind the edges if you didn't fancy doing it yourself.
https://www.woolies-trim.co.uk/c-76-carpet