Lookup does not appear to expose any methods which allow focusing on the underlying TextInput component. focus is a public method exposed by TextInput. Lookup should also expose a focus() method which will allow end-users to focus on a specific Component.
This is preventing us from adding autofocus to a Vue-injected field at T153988: Migrate Special:Nuke to Codex. It's technically possible to use this.$refs.lookupComponent._.refs.textInput.focus() (where lookupComponent is a ref attached to a Lookup), though this is a completely inelegant solution and involves accessing the target object of the Proxy returned by this.$refs (which should be private).