If your Magento 2 Blog search does not work, most likely the issue is in your theme that overrides original blog templates and layout files.
Try the next steps:
1. Make sure that you have the latest Magento 2 Blog module version.
2. Try temporary remove Blog layout XML files and template PHTML files from your theme one by one and run static content deploy. Once find the file that brings the issue, pay attention to it.
3. Check if a blog search button in the sidebar has the attribute "disabled", if yes then it is disabled (not active), and this may be a root of the issue. In this case, find a code which makes it disabled. The disabled attribute can be added directly via .phtml file in your theme or with some JavaScript. If you cannot find it, then you can insert this JS script to undisable the button:
<script>
require(['jquery', 'domReady!'], function($) {
setInterval(function(){
$('#blog_search_mini_form .action.search').removeAttr('disabled');
}, 2000);
});
</script>
