My assumption is you got the user interfaced locked down already--meaning you are good with the UI. You just need the live search.
Possible solutions:
1. Client side search.
The table component that you've used must have a in-memory, in-place search capability. If it's a custom table you have created then you will have to implement your own live search.
My assumption is client side search isn't applicable for you otherwise you wont be asking this question.
2. Backend search via Sql queries
You can implement a backend search using your Sql database's fulltext queries. Google it on how to do text queries. Of course you will need to create the backend service to handle this logic so that the UI can call it.
3. Backend search via Elastic Search
This one gives you the best possibilities of all the three solutions but it is more involved. First make sure you know how to run ES. Make sure you know how to create mappings and indices. Check the ES manual or yôutubê. It is not that difficult. It looks intimidating at first but it is not.
With ES you can do queries like Google's (on a lower scale of course) and also have auto complete feature when making a search (but this one is really combination of ES and AJAX).