Custom 'flex copper' ribbon cable
What you're talking about is a flex(ible) circuit. They are often made by different factories from rigid board, perhaps with the same corporate name. 

They are available with various numbers of layers, thicknesses of copper, features such as stiffeners and finishes and other options. Often they use orange polyimide (Kapton is one trade name) for the polymer substrate. 

Shopping recommendations are off-topic, but you can find many sources for flex circuits. You use more-or-less the same CAE tools to design them, but NRE costs tend to be higher. Prototyping thus tends to be expensive.