It depends where you are, where it's going, and what bumper. A 69 Charger and 70 Charger are two very different dimensions and different in weight. A bumper is pretty strong, so one can cut down a bicycle box to build for a daily driver. A newly chromed bumper deserves a minimum of 2 inches in all directions. Anything less is not insurable. A bigger box should be made with refrigerator or wash machine cardboard. It is like a weak plywood. Just guessing, the 69 bumper will ship UPS or Fedex, but a 70 may have too large of dimensions for them: 130 inches total is the limit, Length and Girth. Fastenal halted shipping last Spring, don't know it they are back again. If they are, prices are great, but they want a crate built from wood which is expensive. I shipped a 57 Chevy hood to Riverside on Greyhound, but since then, they have cut their allowable dimensions and raised their prices a lot. They were competitive with Fedex and UPS, and shipped a box that was nearly 24x48x96, but no more. Also, your box gets bumped for passenger luggage, but buses run 24/7, so things move quickly. The Chevy hood took 48 hours to go 1,500 miles. Same with the Honda XL75 I shipped to R.I.
Depending where you live, there may be localized shippers who ship to your neighboring States. Then there is uShip which is a reverse auction where shippers bid against each other for your job. Look for owner/operators and avoid brokers as they add a layer of communications and fees.