You have several issues to deal with.
As a tax matter, you are using a personal cash account for someone else's business. The money will not be taxable to you, but you will need to keep meticulous records to show the IRS if you are audited. I believe Venmo will be required to issue a 1099-K to you if there are more than $20,000 of transactions per year. You would leave it off your tax return, file by mail with a written explanation instead of e-filing, and be prepared to show the IRS your extensive and meticulous proof that you paid out everything that was paid in and none of the money went to you personally. It would be far better for the organization to have an account in its own name, but that may not be practical.
There are several other issues you are exposed to as a matter of tax compliance and good governing principles for your organization.
1. You should not be the only person reviewing these transactions. Have another officer from the organization review the Venmo transactions and verify that the amounts paid to you by Venmo equal the amounts you paid out (whether you paid for supplies or whether you paid the organization and the organization paid for the items.) Since you are managing the Venmo account and you are also the Treasurer. you could very easily be falsely accused of embezzlement by someone with an axe to grind. Have someone else review and sign off on all payment that come to and from your accounts.
2. Related to that, it would be best practice for you to collect the Venmo payments and write ONE personal check to the organization for all the Venmo payments, then use the organization's checking account to pay your vendors (t-shirts, party supplies, and whatever.) Don't make those payments from your personal account if you can avoid it. Keep your personal financial ties as simple and transparent as possible.
3. Even if some payments made to your Venmo account would normally represent tax-deductible contributions to a non-profit, they are NEVER tax-deductible if made to a specific individual (meaning you). You must not issue receipts that say the payment is tax-deductible, or else you personally and the organization can get in trouble. Only contributions made directly to the organization are tax-deductible, and only then if your organization follows all the other applicable rules. If fact, you probably shouldn't issue any tax donation receipts unless you thoroughly understand all those rules.
4. It sounds like you aren't really a separate exempt organization, you are operating under the umbrella of the school. If you were a separate exempt organization, the organization would have its own tax number and be able to open a bank account or Venmo account. That doesn't really change anything I wrote above, except that you really can't issue tax-deductible donation receipts if you aren't an authorized representative of the actual EO. If the club is a separate EO with a separate IRS registration and tax number, then you really should be able to open accounts in the EO's name. The school can't really stop you if you have your own separate EO registration.
