A simple test of the REST calls worked fine, but I wanted something more appealing.
Luckily, Vuetify has some nice Material Design components ready for Vue and also a nice autocomplete component.
This method watches the query input field and uses the REST backend to fill the results. I used a delayed promise there to allow the user to input the complete search text before I use the text to search the backend.
Selecting a result would trigger a location change to the actual terraform documentation page.
The hard thing here was to get my head around how Vue plugins work, how they work in Typescript and how Vuetify components and their specialities (like slots) work. But the good examples and documentation for Vuetify and Vue helped a great deal.