What is the best API for Search feature on website?

The best API depends entirely on your data source, your budget, and your needs.

If you are looking for a sophisticated solution, try out:

I use Elasticsearch… It's super fast

Any specific reason to choose it over Agolia?

no idea, Ruby people use elasticsearch, that's how I learned it

You can build your own with MongoDB or PostgreSQL.

Also elastic search from AWS is amazing.

If you don’t want to pay before testing some elastic search services from Heroku are free.