Yes, for bulky items, Greyhound ships reasonable but slow, since your bumper may be removed from a bus to fit luggage, then placed back on another bus.
I had a large front grill shipped across the country. Was not expensive at all. But was held over a few days in St Louis.
Best to arrange drop off and pick up from Greyhound bus depots and go pick it up when they contact you.
If your far from a decent sized Greyhound facility, then you have the travel to get to it.
-Also, Fastenal ships quicker, anything bulky like engines, trans, bumpers, body panels... but only if it can be secured to a pallet. If it's too large and can't be secured to a wooden pallet, they don't ship that, because they need to be able to fork lift pallets around. (via Blue Lane Freight).
Check them out.