Can I request a 30-day extension while waiting for my school to prepare documents for my visa extension in Thailand?

Nope, as

@Ellie ******

said, there are no 30 day extensions for people on extensions based on employment. You either get the documents in time to apply for your extension BEFORE your current one runs out OR you end up on overstay <- If you are going to get the documents in just a few days after your current extension expires you can just overstay a few days, pay the 500 baht a day fine and then get your extension like normal,.

You definitely need to talk to your employer about this because in Chiang Mai you can apply for an extension when you have 45 days left on your current one. There is NO reason they ran the clock out on you without giving you paperwork for your extension