As others have said, we can’t really help you with overcoming the unfairness of the vote (aside from suggesting a different method for choosing which teams go and/or explaining why your robot is the best to the teacher/voters). However there are some great resources on the forums to combat the underlying funding issue you described.
Here is a link to a topic with some great resources for getting funding for your team/ school teams: Guide to Getting Sponsors
This isn’t the only such topic on the forums so I encourage you to search for keywords like “funding” and “sponsor” in the forums’ search tool.
Good luck!