I checked a few EU embassies sites for requirements, none of them explicitly says that you will have to be a permanent resident to apply for the visa. While I am sure on other EU embassies (such as in UAE) you are required to be a citizen or a permanent resident.
Anyway, I think it is possible as many travelers especially ones who travel for long times do apply for tourist visas on the road. I also think you will face some problems though, one of them is providing a recent original copy of your employment certificate and a recent bank account statement.
Your best bet is letting someone back home email you these documents and visit a consulate with all the documents and then explain to them your situation, I guess you will get it. Remember, your stay and intend to leave the US is a positive thing and I guess it will help you in getting the Schengen visa.