It's not always absolutely necessary to hire a shop that only makes screws, if that's what you need. Many times, a general machine shop can do the work if they have screw machines. Just do a little researching and ask if your shop will manufacture screws.
The two versions of screw machines are multi spindle and single spindle. If you know what machines you need, you can ask the right questions and find the right company.
Other than having the right equipment, the most important thing that the company must have, is quality control. The manufacturing must be precise, efficient, and consistent.
The material used to make screws can vary. The most common are: stainless steel, copper, carbon steel, aluminum, and plastic. Speak with the machine shop reps regarding the material that would be best suited for your design and use. Cost is also a factor, and each material has it's own price point. This may be a place where you can maximize your savings.
It's a good idea to shop around for the right machine shop to manufacture your screws. It would behoove you to get a least four quotes before making your final decision.