Zen Cart can handle this for you beautifully, right out of the box!

Simply log into your Zen Cart admin, mouse over the Configuration menu on the far left, then go down to Customer Details.

Once there, look down until you see Customer Approval Status.  Click to edit, then select the option that best serves your business.

You may also just need to select #1, Must Be Authorized to Browse - depending upon what your needs and goals are.