I am interested in sewing a raincoat that looks a bit old fashioned- rubbery, shiny blue. The closest thing I could find was the image I've included. My problem is that I have no idea where to get fabric like this. I've looked locally and online, and I haven't really found exactly what I'm looking for. Any advice?
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:
It's likely you'll find other resources as well.
That is Napa leather, its waterproof like oil cloth, but thinner and ideal for clothes. Napa is sometimes written with 2 P's. Keywords you might want to try to buy it online are:
napa leather, nappa leather, glove leather, sheepskin, artificial leather, synthetic leather, faux leather fabric
If you search "napa leather" on, for example aliexpress dot com, you'll see exactly what you need, and really cheap also. Thickness is important, check that also. I guess less than 0,6mm might be what you need, but it's so inexpensive that you could buy a bit of several different thickness values, and find out what's best for your needs. Good luck!