Is it possible to build a full Node.js web application using Kotlin?

Yes, it's possible https://kotlinlang.org/docs/reference/js-project-setup.html

But, NIO was the biggest reason to use NodeJS instead of any language to build a backend solution. Now, with reactive first class support you can have a stack like Kotlin + Spring Reactive + Coroutines + R2DBC and build a simple micro service or any full enterprise solution.

Technically speaking, yes, provided the claim by Kotlin that:

You can use Kotlin to interact with server-side JavaScript such as node.js

Is correct, and the transpilation of Kotlin -> JS is reliable enough to be able to predict what JS is coming out, then you could write a Node app in Kotlin, much as you can write them in TypeScript.

I suspect, personally, that you'd find it difficult, buggy, and rather short on support, but it might make a good academic exercise...maybe.