User is trying to create purchase orders from the Budgets > Create Purchase Order screen. However, the Vendor ID drop-down shows up blank. When the user is creating purchase orders from the Payables > Purchase Orders screen, vendors show up. So, why are they not appearing in the Budgets screen?

The reason vendors don't appear in the drop-down on the Budgets screen is because they have to be assigned to their respective budgets first. After you assign them, they appear in the drop-down.