Depends on if you're an immigrant or non-immigrant alien. AFAIK, there's no law that forbids immigrant aliens from possessing firearms, but non-immigrant aliens can only posess one if they're in possession of a hunting license or permit lawfully issued in the United States (See Title 18, USC Chapter 44, Section 922, part (y)(2) for details.)
Issues of federal law aside, If you drive Route 66, you'll be passing through many states, two of which are Illinois and California, which have some of the most restrictive gun laws in the U.S. You will certainly not be legally allowed to pack heat in either of those states, either openly or concealed, and wouldn't be even if you were a U.S. citizen. Then there's Oklahoma, which allows no carry whatsoever without a permit, although permits are shall-issue and easy to get.
Needless to say, it will be a big pain in the backside. However, If you really, really have your heart set on doing this, and are willing to go to all the trouble, here's what I'd do: Establish residency in Arizona and get yourself an Arizona carry permit. That will allow you to legally carry in every state you'll pass through except California, Illinois and New Mexico. New Mexico won't be an issue, though, as long as you bring a belt holster, since New Mexico only requires a permit to carry concealed and then only while on foot - no permit is required in New Mexico to carry either openly or concealed while in a private vehicle. Ironically, your Arizona permit won't be needed in Arizona, we don't require carry permits and only issue them for purposes of reciprocity. California and Illinois are at the ass-ends of Route 66, so that will allow you to drive the vast majority of the road while packing.
Disclaimer: I am not a lawyer, and the jails in the U.S. are filled with people who took advice from people who are not lawyers. Research carefully before you decide to try this. You might even want to want to consult a U.S. lawyer before you try it. It will cost you about $100 USD, but that's still far cheaper than a felony conviction. If you decide to consult a lawyer, I recommend this guy:
He's about as pro-second Amendment as lawyers come.