What materials can I source to make a shiny/rubbery blue raincoat?
It's strongly likely that you are seeing what I know to be oilcloth. Per the link, it is close-woven cotton duck or linen cloth with a coating of boiled linseed oil to make it waterproof.

Using oilcloth as the search term, I found a few locations online for purchase. As your post referenced the blue color, I focused on that specification:

Oilclothbytheyard.com

It's likely you'll find other resources as well.