Hello again and thank you for the additional information.
In a sole proprietor situation, the individual owner is the business so there would typically be no difference between signing a contract as an individual or in the company name. There is no liability protection provided sole proprietors who have not incorporated normally, so the owner would be personally liable for any debt incurred by the company.
To sign a contract in the company name, the contract could use the company name as one of the parties entering into the contract, then the signature line would typically include "your name, owner".
As for the vendor registration, some government entities permit companies to register as a vendor for that entity, although I am unsure what the individual meant by a registering as a bank vendor. Normally, organizations do not have a formal registration process to register as a vendor except in some situations where they might be working on a government contract. In that case, you would typically contact the government contractor and inquire about registering as a vendor to provide services.
Here is a link to the L.A. County vendor registration page, which provides an example of registering as a vendor:
I hope this helps clarify the situation for you. Please remember to rate my service once you have all the information you need. If you have any other questions, please ask me – I’ll be happy to respond. Thank you!