Hello,
That widget comes without Ajax Search but our Elementor Search element has an option for the Ajax search and you can show the Static block with that element in the Main sidebar. Check it now.
Added custom css to make ajax search results showing without issues
.shop-filters-area:before,
.shop-filters-area:after {
content: '';
display: block;
clear: both;
}
.template-content .shop-filters {
overflow: visible;
}
Regards