There are two ways to require your customer to enter additional information:
1) Create a form (Site > Forms) and attach it to the checkout page in Site > Settings > "General Info" tab > "Checkout Forms" section. When you create the form, make sure to mark the field(s) Required.
2) If your customer will not be using purchase order codes, you can use the Purchase Order feature to require the user to enter additional information. In Site > Settings > "General Info" tab > "Checkout" section, check the "Allow custom purchase order code" box, and select "Required" from the Purchase Order Requirement dropdown menu. In the "Purchase Order Prompt" box, enter a prompt telling the customer what information they need to enter.