Is there a way to post PDF rather than send?

I don’t think this would fit with Contact Form 7 integration unless you can find a ‘download attachment on form submission’ type plugin. However, I have plans to write another plugin that would do what you are describing.

For now, what you can do is the following,

1. Create a button that would run a JavaScript function.
2. Create a JavaScript function that would take your form data and submit it to /pdf-fill.php as a POST request with array ‘data’ argument where the keys would match the field names in the PDF form
3. Create php-fill.php in your public_html with the following code,

<?php

define('WP_USE_THEMES', false);
require(dirname(__FILE__).'/wp-load.php');

$data = $_POST['data'];

// change this when using a different PDF file
$attachment_id = 123;

try
{
	if( !class_exists( 'WPCF7_Pdf_Forms' ) )
		throw new Exception(__("WPCF7_Pdf_Forms not loaded"));
	
	$tempfile = tempnam(get_temp_dir(), "pdf_");
	if(!$tempfile)
		throw new Exception(__("Failed to create a temporary file"));
	
	$service = WPCF7_Pdf_Forms::get_instance()->get_service();
	if(!$service)
		throw new Exception(__("Failed to get PDF forms filling service"));
	
	$service->api_fill( $tempfile, $attachment_id, $data );
	
	$file = fopen($tempfile,'rb');
	if(!$file)
		throw new Exception(__("Failed to read a temporary file"));
	
	header("Content-Type: application/pdf");
	header("Content-Length: " . filesize($tempfile));
	
	fpassthru($file);
	fclose($file);
	@unlink($tempfile);
}
catch(Exception $e)
{
	@unlink($tempfile);
	header($_SERVER['SERVER_PROTOCOL']." 500 Internal Server Error", true, 500);
	print $e->getMessage();
}

This should do what you are looking for.