You need a couple of things in place. You need to have your webserver that implements SQL-databases also or your server needs to recognize that you can call an outside SQL-database querying program, that you may have to write yourself. I feel you want to be able to implement your own webserver and SQL-database management and querying program.

The HTML you write will be server-specific, meaning that it will not work with other servers that don't implement the same functionality using the same naming conventions. The Internet was designed to be personal when it comes to your homepage and having to make use of an external server as such is a drawback, because your are forced to work with their systems.

The HTML-conventions are really simple in terms of your form. You make use of a POST-request (METHOD="POST") to send data and with the ACTION-attribute, you either send your form data directly to the database management system, specifically the querying application, in the same way that you would send it to a PHP-page, or to the function-name that the server specifies you should use for SQL.

In case of a to the server external application you designed, your server has to be set up so it recognizes this application and the application needs to send back HTML to the client connecting to your server to display the result.

With elaborate search forms, you could opt to parse the query using JavaScript so you have an SQL-query. That way your server-side database querying application can be kept minimal in that it only has to interpret SQL queries. This way you can use it for different database structures custom-designed for every webpage you mean to use a database for, instead of every webpage needing a new application for every new specific database architecture, that parses the query rather than making use of javascript.

If you are not going to implement your own server and database system, your answer lies not in the many API's but in searching for a webserver that also implements SQL-functionality with a clear manual on how to use it. You really don't want an additional layer between your HTML, JavaScript, and the database. Quite frankly, you need to know exactly what your server does and with all of these external systems, who knows who ends up with your data?