While Search UI doesn't have a component library for Vue.js, you can still use them together, just with a little extra work.
To achieve that you'll need to import and work directly with Search UI Core API. Read a short Architecture guide for high-level overview on the approach.
Once you're familiar with the concept of Search UI core, the following docs will give you deeper dive into available APIs:
The following demo is a good starting point for your Search UI implementation. The source code is available on GitHub: https://github.com/elastic/search-ui/tree/master/examples/vue