How can I add a zip code service availability checker in WordPress without Woocommerce? [closed]

This is much simpler version of your question:

Given a list of strings, how do I check the user input is in that list?

You don’t need a special service or fancy software to do that, just make a big list of the zipcodes you support and use in_array.

const $supported_zipcodes = [
    'zipcode1...',
    'zipcode2...',
    ... etc
];

function ncblender_check_zipcode( string $zipcode ) : bool {
    // clean up the zip code, remove trailing spaces and make it all lowercase
    $zipcode = trim( strtolower( $zipcode ) );

    // don't do empty zipcodes
    if ( empty( $zipcode ) ) {
        return false;
    }

    // if our cleaned up zipcode is in the list, return true
    return in_array( $zipcode, $supported_zipcodes, false )
}

Now you can create a generic form with a text input, and use that function to check if it’s a supported zipcode or not. You might want to find better zipcode validation code online as what I put above is a simple/crude check, but that might just be enough for you