Hit up the following places with a well written sponsorship letter or call them up and see if you can arrange a demonstration for them.
http://www.ekaassociates.com/
http://www.gleicher.com/
http://www.ngkengineering.com/
http://www.ltk.com/
http://redcomllc.com/
http://www.starcomfiber.com/
That was through a simple google maps search of "Engineering near Scotch Plains, NJ"
Edit: actually, your city has made this very simple for you -
http://www.visitscotchplains.com/members.html
Just make sure you write up a great sponsorship letter. Many examples are here in CD Media if you need examples. Check out this one that I helped a team with -
http://team2495.com/index.php?page=donate
If you spend time researching, you can find plenty more place to contact. Visit doctors, dentists, law offices..etc..
Couple this with doing demo's in front of grocery stores and supermarkets as Adam suggested. By simply standing outside one weekend with your robot and talking to people about the team, you can rake in $400-$800 at a decently busy store.
Shoot me an email or PM if you guys need more direct help. Your teacher is doing the right thing by saying the team should be raising more money otherwise should participate in a different event. Rise to the challenge your teacher is presenting you. Find more mentors, find more money, I am pretty confident you can do it.