Is there a PHP library or API that would let me quickly build survey questionnaires and dump the captured data to MySQL?

If you are familiar with LAMP and PHP, why not build a little CRUD application?

I’m have the guts of one now, but it’s a pain to update the schema every time, branding it is a drag because I didn’t lay out the CSS properly, etc, etc. I’m hoping there’s something off the shelf I can play with quickly, since this seems to be coming up a lot, and I don’t have time to build something flexible enough.

So you are looking for an open-source survey application written in PHP for MySQL.

There are tons of them on www.sourceforge.net

Awesome. Which one is good?

Up to you to try. I write my own applications. Look for something with active development.

check out Advanced Survey (www.proxy2.de) and LimeSurvey (limesurvey.org).

Lime is very professional for an open source project and they recently released a significant upgrade.

www.cakephp.org

I’ve been able to set up an application like you describe in five minutes.