If you go to the Atlassian Marketplace (https://marketplace.atlassian.com/search?q=notification) and look at the notification plugins, that is a great place to start. I'm not sure if any of those plugins are an exact match for what you are looking for but it might be worth starting there just to keep from reinventing the wheel.
In the past, before I joined Atlassian, I worked for a startup that had survey emails go out from JIRA. When the ticket was closed, we had JIRA send a notification to a middleware service I build that queued up the emails for a specific time period then forwarded them to an email delivery service (e.g. Mandrill or MailChimp) and from there to the customers.
You could probably get away with building a plugin for your specific needs if you wanted to as well. I would probably go for a Connect Plugin as opposed to the P2 Plugins, though. Connect gives you the ability to work in most any language (not just Java) and to build a service outside your JIRA instance for these surveys. Had I known about Connect before joining Atlassian, I would have used that technology to do mine.
Hopefully this helps.
--Sam
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