Ok here is a direct answer.
No!
Is that enough or do you need an explaination
The PTO/PTA is a seperate organization. It is NOT a part of the department of education, the school board, or even the school. The school board has NO authority over it, except to the extent that the PTO uses school facilities under the rules established for such use for ANY private organization (read - boy scouts, girl scouts, booster clubs, kiwanias, etc).
An Audit is a very specific and legal tool. The only agencies that have the power to audit you are tax agencies and state and federal prosecutors. If you are non profit (501(c)(3)) you are required to make certain information available to "individuals" that ask to see it concerning your financials, but you do not have to answer their questions, you do the tax mans!
This means if your school board trustee wants to see your books as a private citizen then she can, provided you are 501(c)(3) if you arent then you need to check your state laws on this, but I bet she is not entitled to see them. As the school board trustee and demanding your books to audit them in that capacity tell her to take a number and you will get back to her. Seriously tell her that you will send her a copy of your annual audit at the end of the business year as a curtosy but her request that the school board be permitted to audit the financials of the PTO has been disappoved as they are not an oversite agency for the PTO.
Justme, someone is overstepping their authority massively! You need to contact all the other PTO/PTAs in your area and get this under control. If you allow this to stand you will be subjecting the PTO/PTAs to government intrusion beyond their authority. Also involve your state representative. As a first step I would send a letter to who ever told you this asking to cite the legal authority for requiring this and asking for the authorization to deny PTOs access to schools who do not comply. Next, take this to the next school board meeting and make a lot of noise. The argument will run along the lines that the PTOs are not auditted and therefore it is our responsibility to do so. Ask them when they will be auditing JC Penny'sm WalMart, K-Mart, Bloomingdales and other private business' since they donate to schools across the country. Also ask who is going to pay for these "audits" of private agencies. You may wish to go in armed with some data on how much it costs to perform an audit times the number of school PTO/PTAs that will be subject to this - also the amount of time and effort it takes to do this - would that money not be better used to provide a tutoring program or othr activity that actually has an educational value? BTW I have not checked on this in a while but the average to do an audit of a small business by a government agency was somewhere around $1150 last time I looked and that is if there are no problems at all found and it takes about 50 man hours. Sounds like someone is trying to increase their staff to me. OH and invite the press to that board meeting and give them a heads up on what will be discussed, they love school board contraversy since they get so little.