This is not a problem with the carriers, border (CBP), or TSA. The industry standard is that the order of the names is irrelevant, e.g. John Smith vs. Smith John is fine. What becomes problematic is when there are misspellings or the names are different.

For travel, you will be required to enter your passport data and may already have done so. Name order will matter there. Make certain that you don't mess that up.