Your issues may be larger than a mere bank account. If the PTO is seen as an independent entity (as is most common), and not under the school's umbrella, then it needs to examine its business structure.
The general rule of the thumb is that if an organization has less than $5000 per year GROSS income, it can operate pretty informally. But if the PTO is exceeding that threshold, it should be set up with the IRS as a 501(c)(3) organization. (The $5000 represents everything coming in; not what's left after expenses.)
Whether the PTO is operating properly (legally) may have an impact on your decisions regarding the bank account.
The PTO should get its own EIN (Employers Identification Number) and establish the bank account. The treasurer and at least one, probably two, other officers or board members should be signers on the account. Usually the President is one of these. (Note - getting an EIN is very simple. No cost; can be done online or over the phone with the IRS in minutes.)
Whether or not the principal should be on the account is a matter of opinion/personal preference and there is no wrong or right answer. Your official role in the PTO may also impact the decision. In our elementary PTO, the principal was a signer, adding valuable continuity and a vested stakeholder who was easy to track down for a signature. (And the principal was one of our board members.)
In our middle school PTO, the principal wouldn't touch our bank account with a ten foot pole. He and the PTO work well together, but he feels that he should have no rights to the PTO bank account.
Here on the Forum, you'll hear both opinions.
Many organizations find it very convenient to have the principal as a signer. Some feel it's a conflict of interest. In some school districts, the policy is that the principal not be on parent group accounts - so that might be a consideration. If you know the PTO is not setup legally, you might not
want your name on the account.
I would think it fairly unusual for the PTO President not to be a signer.
To learn more about properly organizing a PTO in the business sense, you may want to view
this post
.
P.S. If the PTO is not independent and basically functions as a committee under the school's umbrella, you should follow whatever the accounting practices are for the school.