Trigger action dispatch from app-webhook event

Is it possible to implement github-app webhook event handlers using only github-actions, and no custom servers?

For example, if I want to run some script when the app has a new installation. Could we just point the webhook for a github-app back to the github-action manual dispatch api, in order to trigger a custom workflow run for the github-app event?

Or is there another, simpler, way to trigger an action for github-app events?