You can mount a react or a vue app partially into a page, so what you ask is theoretically possible but probably not the best solution.
You may consider expanding the react application a bit and go fully headless. In this case, next.js (based on react) might be a better solution as it will allow you easy static page generation/server side rendering. Of course this will be not as easy as a regular WordPress theme editing job, and definitely will take more time. In any case, I suggest you to use graphql instead of wp rest api which will perform better.
On the other hand, the poor performance may not be because of the ajax calls, but because of the poor query/database performance which react or graphql cannot solve as well.