Intramural elections are exempt
Putting up posters and holding rallies for student council president are not generally considered a violation of the 'no politicking' provisions of the law. Internal or with in the bounds of the school community activities are acceptable. Read Rules for Exempt Organizations During an Election Year for further guidance.
Politicking for local, state and federal issues and candidates forbidden
In an election year where emotions are running high and record numbers of young people are being drawn into the process, you need to be very careful that your school complies with the law. Make sure you state your policy clearly in the school's handbook. Enforce that policy. The last thing you need is for somebody to file a complaint with the IRS and put your tax-exempt status in jeopardy.
Prove your 'no politicking' policy in your Form 990 filing
Schedule A of Form 990 gives you a place to document your 'lobbying' activites or lack thereof. Remember: your school's membership in NAIS and other regional independent school organizations can be construed as 'lobbying'. Generally the amount spent on memberships is relatively insignificant when compared to your overall expenditures. Just be certain to record those memberships and make the appropriate disclaimers. Your tax adviser will guide you through that process.
No matter how tempted you may be to support one candidate or another in any extra-mural (outside the school) election or political activity, don't do it under the school's name or auspices. You could put your school's tax-exempt status at risk.
Disclaimer: the author is not a tax attorney or accountant. Seek appropriate legal and accounting advice.