custom shipping quote before allowing payment

For customers in certain regions, I want to quote shipping manually before allowing them to complete checkout. They would have to contact me, I would do the quote, and they would enter something on checkout, like the quote#, and the system would look up their quote# to confirm that it's valid, retrieve the amount, set the amount as their shipping rate, allow them to continue with checkout, and update the database to indicate that the quote# was used.

Is there anything like this, or another system made to allow for customers needing a custom quote? I currently have customers call me, and I have to take the order over the phone, but that is inefficient for me.