What state can you open business under without giving away our SSN?
Generally the states (with a few exceptions) don't require you to provide them with Social Security Numbers. Your SSN would be needed primarily when obtaining an EIN for your company and if your company is to be elected to be taxed as S-Corporation.
If you have no SSN its usually not a problem and you can just omit the fileds that ask for it (put 000-00-0000). Keep in mind that only US persons (with SSN) can own S-Corporation of have their LLC elected to be taxed as S-Corporation.
If you have SSN but will obtain the EIN by yourself you can do the same with the SSN field. However if you want our help with that we won't be able to obtain it without having your SSN on the SS4 form.