Here's a rough script idea:
Go to Record/Request/Page [First]
If [yourtable::oldcarrierfield = "UPS"]
Else If [yourtable::oldcarrierfield = "FedEx"]
/* And so on and So forth...*/
Go to Record/Request/Page [Next; Exit after last]
rdowler's suggestion is really clear, and that is a big plus. If you want to do it in less (AND you are guaranteed that only one carrier has been ticked across all of the 5 fields - that's also a job for a radio button set, not a check-box, but anyways) then you could amend it to be:
Set Field [yourtable:NewCarrierField = OldCarrierField1 & OldCarrierField2 & OldCarrierField3 & OldCarrierField4 & OldCarrierField5
Go to Record [Next, Exit after last]
Or just use Replace Field Contents with the calculation option, using Sorbsbuster's calculation, but no script.
With all of these approaches, make a back up copy of your file first just in case you don't get the results you expected.
Woohoo! Thanks everyone! This worked perfect.