I would add one more piece of advice: announce the issue one month and vote on it the next (if you can afford to stretch it out that long). We are dealing with a large unbudgeted proposal right now, too. If you vote on it at the same meeting where it is first proposed, then you assume you have all the information your members who happen to attend that month's meeting need in order to make an informaed decision. If you table it for one month, more info can be gathered (ex: teachers could solicit donations from the community, an alternate fundraiser could be set up, etc.) and more minds will be thinking about the best solution for the problem. Next month, revisit the motion, discuss, and vote. Your VP can vote like everyone else.
I wouldn't recommend this approach for every spending request, but if it's potentially controversial or a significant portion of your budget,or if it will require you to reallocate previsously-budgeted $, then you owe it to all your members to make the issue public. This way, you avoid the risk of having a special interest group show up one month and vote to allocate $$ to their pet project (that's not meant as a slam to the teachers, I'm talking generically here).
I would leave it up to the voting members of the PTO and see if they will approve it.
To pay for it, you will need to either remove something from the budget or have another fundraiser to pay for it. While at the meeting the president or treasure will need to explain to everyone that this was not a PTO event, but the funds were requested from the PTO by the teachers. Let them also know or vote how to pay for it. Get the members involved...
Your VP may have issues with this, but it is not wholly his/her decision. As you stated, according to your bylaws it must be presented for a vote to the general membership. It's a request being made for funds, so I would put it to the membership for a discussion and vote. I would however ask the teachers involved if they had plans to seek the funding anywhere else and explain that it would be a big portion of your budget. This way they will understand the issue some members may have with the request and that they shouldn't rely wholly on your group for the funds. If this was my group, that question would be one of the first asked when the request was presented so you will probably need that information for your meeting anyway.
We've had issues that some people got all wound up about, but the fact is the decision was in the hands of our general membership. Everyone got the chance to voice their opinions and concerns, we voted and then moved on.
Long story short:
In the aftermath of hurricane Katrina, our school administration partnered with a school in Mississippi to help re-outfit their school. Donations were sought from families such as books, backpacks etc. Many were received and shipment was sent.
This is all good: fostering philantropy from students, random acts of kindness etc.
This project was run completely by the teachers. It was not a PTO sponsored or sanctioned committee. Implementation was not discussed with us, we had no line item and no committee. All that is fine also as I applaud the teachers owning this venture.
Here is the problem. The teachers are coming to the PTO and demanding that we vote on underwriting the shipping costs--> $900 is being asked for. Anything over $500 must be voted upon by general assembly according to our bylaws. $900 is alot of money in our PTO. Most of our committees have working budgets lower than this.
The teachers did not do any sort of review of alternatives, they simply assumed we would foot the bill after the fact.
We are a 501 C 3 (newly) and our VP is having huge issues with this. She doesn't see this as being a PTO responsibility.