Unfortunately AWS has been severely neglecting meaningful IPv6 support. Their "solution", if you can call it that, is to throw a non-VPC ELB in front of your application. For many reasons that is a non-tenable solution for many use cases.
As a stopgap fix, the tactic I've seen used by several organizations is to stand up an IPv6-capable VPS with another provider, point the
AAAA record there, and then use haproxy or another similar software package to proxy requests to your IPv4 infrastructure. Yes, this is a really awkward and admittedly broken way of doing things, but until AWS releases meaningful IPv6 support, there aren't all that many other options.
Regarding Craig's assertion that all ELBs (VPC or otherwise) have IPv6 addresses. Yes, that is correct, but in my testing, and in discussions with other AWS experts, the ELBs don't actually listen on those IPs for traffic. My guess is that enabling IPv6 on the ELBs as they've done is a first step towards offering full IPv6 support at some point. My suspicion is that they will make an announcement about this during or right before their re:Invent conference the first week of December 2016.