I think @Hal_Al knows something about Ohio state returns. Are you asking if you can attach a document to your return, like if the state requires proof of something? I believe that Turbotax does not allow any attachments, even if the tax service allows attachments. (This was a company decision, as far as I can tell.) If an attachment is absolutely required, you may have to print your return and file it by mail.
If you are really asking if Turbotax can download a document, the only documents I know of that Turbotax can download are W-2s from participating employers and 1099-B, 1099-INT and 1099-DIV from participating banks and brokers.
