They CAN! All they have to do is find a lender who is willing to lend them the money at a rate they agree to repay. Student loan lenders make loans to those who would be unable to get a lender to make that loan because the government has guaranteed the conditions, and if it is a direct or Plus loan through the government, the rates.
Lenders are in business to make money. They usually set their rates, fees, and terms on the repayment history of their customers. If the student loan programs change, the lenders will have to decide if they are willing to make unsecured loans to borrowers (students) without a credit history, and if so, at what rates. I think we’ll find there won’t be many lenders who will do it, and if they do it will be at high rates.