What is the name for a company that contracts with other companies to mass produce products? And how hard and what is the process to get a contract of such for a small startup?
I believe that the companies you are referring to are either industrial design companies or manufacturers.
Industrial design companies usually have the skills to take a concept or a working prototype and design a commercial product. Sometimes they also have the skills to manufacture the product themselves, but many times they can help you choose a manufacturer for your product.
Some manufacturers have teams that help small companies to transform their concept or prototype to a mass produced product.
Usually these companies are service providers, meaning that they will charge you for their time and their materials, and mass production is quite expensive. It means that you would have to have enough capital to finance the mass production. Nevertheless, if you are willing to share your revenues with them, and if they find your product interesting and with market potential they might be willing to partner with you and share the costs or take them on themselves. If they do that, they actually act as investors, and finding a company that is willing to do that might be as difficult as finding an investor.
What you should probably do is to talk to several of these companies, get a few quotes in order to know how much money you need, and then go and find an investor.