Build and deploy a search suggestion endpoint for a search engine chrome extension

I want to deploy a search suggestion endpoint (Like in the attached image). I developed a search engine that display search suggestions: https://konlap.com I also developed a search engine chrome extension that overrides Google´s search engine and uses mine instead. So, I want to build a fully featured search engine and want to add these search suggestions in the search bar. Please write a simple explanation on how you would build this behaviour.