This is a fantastic idea, and we do have a PTA.
Unfortunately, I will be unable to present it to them, as all PTA meetings are during the business day (around 2-3 PM) and direct contact info for PTA leadership has not been distributed to the rest of the parents.
We have a school directory that includes the whole elementary school (that parents can "opt out of" if they don't want their info in there) that we pay about $7 each year, but it usually doesn't come out until November.
Because of that, every year the teachers have sent home a form on the first day of school that they will come up with a class list (once again, giving parents the ability to opt out) within the first week of school and send it home (for free).
My oldest just moved to the middle school (where the individual class listings aren't offered) and I still have 2 kids in the elementary school. Before this year, only about half of the teachers that we have had have offered this "service". This year they passed the rule that NO invitations (even if you are inviting the whole class or just boys/girls) may be distributed in school...so every single teacher is offering it to their classes.
Can you talk to the teachers and ask if that can be done? Or even ask her that, if you make up the form and agree to type out the "class directory, could she just pass out the forms to the students and have them returned to you so that you can make the list?
** Edited to add: and have you checked the website for the school for the PTA emails? All of the Home & School Board members have ALWAYS been listed on our school website!