an EID (employer Identification Number) is not necessarily required. If you are a sole proprietor you should be able to use your social security number. You will need to get a state sales tax certificate in order to pay your taxes and to qualify to buy wholesale tax free. You may also need a business license depending on the state and local.
Any site that claims to be a wholesale site and doesn t ask you for a tax id is a FAKE stay away from them they are middleman.
Yes you need a retail sales tax number. Wholesalers cannot sell to you without it.