I've never tried VersionEye. They do seem to have the most comprehensive language support of the tools I've tried, which is awesome. But am I correct in understanding they don't support web hooks for running dependency checks when you push to a GitHub repo?

We are currently working on exactly that. The idea is that we check your dependencies every time you push to GitHub. Here is the current status of that hook: https://github.com/github/github-services/pull/727. I hope to get it done next week.

Excellent! Looking forward to it :) I think the time-based checks are still relevant as well. The way I see it, Webhooks track added or changed dependencies during active development and the time-based checks make sure the project doesn't get stale.