I think it would be like this:

Let us assume that the ID # for our form is 12345, that the essential grid you have made has an alias name of my-essential-grid, and its ID # is 67890.

On the page itself you might have code like this to output our form and the essential grid:

[searchandfilter id="12345"]
[ess_grid alias="my-essential-grid"]

And below would the code added to your child theme functions.php file:

add_filter('essgrid_query_caching', 'eg_stop_caching', 10, 2);
add_filter('essgrid_get_posts', 'eg_mod_query', 10, 2);
 
// turn off caching for your grid
function eg_stop_caching($do_cache, $grid_id) {
	if($grid_id == 67890) return false;
	return true;
}
 
function eg_mod_query($query, $grid_id){
	if($grid_id == 67890) {
	//modify the essential grid query arguments
	$query['search_filter_id'] = 12345; 
	}
	return $query;
}

In the form, the Ajax Container would likely need to be something like #esg-grid-67890-1-wrap. But … Essential Grid does a lot of changing of classes when Ajax is fired, so it is best to leave Ajax OFF for now, sorry.

And the form set to Custom Display Results Method.

Does that help? I have never done this myself, but this is my understanding of how to do it.