Depends on how are you using Vue.js. Are you doing Kotlin/JS with it, so that webpack is run by Gradle, or is it external to your Gradle project containing backend code?
For the first one you'll have to configure dependencies on two subprojects, so that building backend causes building frontend first, and then you collect built JS files to serve them when server is launched. You should have something similar to this:
https://kotlinlang.slack.com/archives/C0B8L3U69/p1575051416301000?thread_ts=1575048475.300900&cid=C0B8L3U69
Unfortunately no out-of-the-box solution which I'm aware of.
If your Vue.js app is completely separate, just grab built frontend manually and put it into resources of your backend app or somewhere in the filesystem where backend would run, depending on your use case.
After you make your frontend files available to backend, simply serve them as static content.
https://ktor.io/servers/features/static-content.html