A business license is required in almost every state. These are normally acquired through a county office. The cost is normally anywhere from $20 – $50 renewable annually. If you operate without a business license the county or state can fine you. You will need to post your license in a prominent place. If you operate from home you still need the license.
Most states also require you to have a Sales Tax or Resellers Certificate. This does two things for you. Number one, it lets you purchase things that you are going to resell sales tax free. This applies only to materials that will go into your signs and be resold to a customer. Your computer or vinyl cutter does not count and is taxable. The second thing the certificate does is allow you to collect sales tax from the customer and remit it to the state. Sales tax will apply to all customers except those who are tax exempt for some reason. If a customer is tax exempt they will provide you a copy of their certificate, which you need to keep on file. A sales tax or resellers certificate is not something that you want to try to bypass. If you are required to collect sales tax then you need to have it. The penalty is more severe than not having a business license. Also, a sales tax certificate normally does not cost anything.
As far as other permits go I do not know of any additional ones that a basic sign shop must have. If you are a corporation then you will have a federal tax id number. If you hire employees you will have an EIN number which is often the same number as the federal tax id.