How can I send information gathered from my customer to the vendor dashboard?

You can use a checkout form with either drop down list with the list of choices (such as a cost center number, department code, etc.) or you can use a single text box area for the users to enter information. You can make it mandatory for the users to select to add the detail. A form can be added to a product page, or Checkout page of your website. The information in the form will pass over to the Vendor Dashboard.
For more details on Vendors, please review this article: Ch. 025. Vendors