Try babysitting. Post notices in the library and online to find clients. Also ask other kids if they have jobs like that and if they know anyone who needs help. You can also do other things like dogwalking: go to a local pet store and post a flyer.
I also suggest you consider selling the stuff you have, and helping other people sell their stuff.
Also go to www.benefits.gov and see if you can get your mother some help. Go to a local food pantry and see about getting groceries. Ask your mom that if you can get her some help, if she can free up some money if she'll share it with you.
Ask your teachers if there are any local nonprofits that help with going to prom.
The survey thing doesn't work.
Help you how? Either you find a legal way to earn some cash, or you learn a lesson: if you can't afford something, you go without.
Sorry but it seems like you are not going to prom. Be thankful that your juvenile criminal record will not be available once you turn 18.
People want things. Get over it.