I would take the check back to the university and have them cut it correctly. Seriously.
However, failing that, I would endorse it in your capacity as treasurer for the company. Your should know how to do that, since you are a treasurer for a company.
Oh. It's an e-transfer.
Absolutely not. You need to go back to the payer and tell them to reverse the transaction and send it to your corporation. A corporation is a separate legal person in tax law, so They literally paid the wrong person.
You can't receive a payment for your corporation. Proprietorship, yes. LLC, maybe. Corporation, nuh-uh.
I'm going to accept it personally anyway
You will get nailed by this. You will get a "1099" issued to you personally, and the IRS will expect personal income tax to be paid on that. You can talk them out of it - I have - but your documentation will need to be tip top, hence the advice for the letter. You must say that the payment was made to you in error and you must transfer the full $10,000.
Also in some instances such payments are subject to backup withholding e.g. That 30% you mentioned. Withholding is not tax. Withholding is a bank account on deposit with the IRS, to assure you have deposited enough money to pay your taxes on April 15. It's technically your money and you get any extra back after you file your taxes.
So if they only sent you $7000, you still have to transfer the full $10,000 to the corporation. And if that creates cash flow problems for you, tough beans: IRS does not recognize the existence of cash flow problems. You might try having the corporation loan you the amount you are shy, but the corporation will need to put all this on paper, set an appropriate interest rate, etc.
If you don't transfer the full $10,000, or if your argument of erroneous payment is not effective, then it becomes two things: a) a prize award to you personally for $10,000, on which you owe personal income tax; and a $whatever investment, for whch you should receive shares of stock. You could also characterize it as a loan to the company.
You kinda see where this "accept it into my personal bank account" option sucks.
I get where passively accepting this is easier right now, I don't think it'll stay easier.
What were the prize rules?
There's one last snag. You have to review the contest rules. Are they allowed to pay a third party? Your corporation is not you. It is quite possible that A) only individuals (natural persons) can enter the contest, e.g. College students, e.g your corporation is not a student of that school..... and B) rules require the prize be paid to the actual winner rather than some other entity according to winner's instructions. (although that last would be pretty stupid, you should be the right to redirect your prize to charity.)
There may also be a tax issue with this. In principle I don't see a problem, the money is income to the corporation and it will pay tax on it via its annual taxes (Form 1120). But I am not a tax attorney.