The Pennsylvania Department of State has a searchable computer database of corporations formed in Pennsylvania. If you tell the company that you have formed a corporation, the company can check the name of your alleged corporation in that database.
I'm puzzled by your question in that seldom does a company care whether an individual is a sole proprietor or a corporation; in fact, generally speaking, if someone is a sole proprietor it's better for those with whom the sole proprietor does business as your liability for errors or violations is unlimited (as a sole proprietor, if you're sued due to a business issue, your opponent can go after your personal property and real estate).
What is the reason the company wants you to incorporate? Would your incorporating be a term specified in the contract you would sign? If not, you have no obligation to incorporate, as a written agreement later in time than an oral understanding will generally supersede any oral terms not included in the agreement.
Of course, as with all of my online answers, my advice is limited by the brevity of your question and the facts provided. Additional information would be required to provide definitive legal advice, so this answer isn't intended to, and does not, create an attorney-client relationship.
My experience differs from Ms. Tamanini's. I have worked with several sole proprietors who have been requested or required to form corporations in order to contract with other companies. Sometimes a company wants to do work with only companies and not individuals because they want there to be no doubt that the individual cannot be considered an employee for purposes of unemployment compensation, workmen's compensation, tax or similar concerns.
If they don't check, you still have issues. If they want to contract with a corporation, the contract would likely name your corporation as the party to the contract, not you individually. Likewise, checks would likely be made out to the corporate entity, not you individually.
Additionally, it is very easy to verify if there's a corporation by searching the Pennsylvania Department of State's Corporation Division's searchable database. Many attorneys who write contracts will do this to verify the party to the contract.
I would have a frank conversation about why they want you to incorporate, to see if some compromise can be reached.
Let me add another cautionary word: you are only permitted to use the words "Corporation" or the abbreviation "Inc." if you are actually incorporated; to do otherwise would be considered fraudulent. So if you would use either of those in an agreement with another company without incorporating, you could have bigger problems than merely whether the other party is checking up on you.