Once a business owner submits the Become a Vendor from the front end, you can then check his application and all other vendor application in Users > Vendor Application as it appears in the following image. Hovering over the application, you can approve or decline an application. You can also view the application for extra fields that has been added by business owner.
After a Vendor Application is accepted, a New Tab will appear on the Business called Products. If the Products tab did not appear for Vendor Businesses, then make sure to Navigate to Wyzi Toolkit > Woocommerce Settings > Display Vendor products Option is ON.