Simple html dom file_get_html not working - is there any workaround?

You must set "allow_url_fopen" as TRUE in "php.ini" to allow accessing files via HTTP or FTP.
Some hosting venders disable PHP's "allow_url_fopen" flag for security issues.

$post = curl_init(); 
curl_setopt($post, CURLOPT_SSL_VERIFYPEER, FALSE);
curl_setopt($post, CURLOPT_AUTOREFERER, TRUE);
curl_setopt($post, CURLOPT_HEADER, 0);
curl_setopt($post,CURLOPT_RETURNTRANSFER, true);
curl_setopt($post,CURLOPT_URL,$website);
curl_setopt($post,CURLOPT_POST,1);
curl_setopt($post,CURLOPT_POSTFIELDS,"regno=$Number");
curl_setopt($post, CURLOPT_FOLLOWLOCATION, True);
curl_getinfo($post, CURLINFO_HTTP_CODE);
$curlresponse = curl_exec($post);
curl_close($post);  
$dom = new DOMDocument();
$dom->loadHTML($curlresponse);

DOMDocument::loadHTML() [domdocument.loadhtml]: htmlParseStartTag: misplaced THIS IS URL : http://www.annauniv.edu/cgi-bin/result/cgrade.pl?regno=11210104001

remove the semicolon from php.ini and restart Apache server to enable php module configuration

; Windows Extensions
...
;extension=php_openssl.dll
...

As I said, your example is working fine for me... But try this way using curl instead:

//base url
$base = 'https://play.google.com/store/apps';

$curl = curl_init();
curl_setopt($curl, CURLOPT_SSL_VERIFYPEER, FALSE);
curl_setopt($curl, CURLOPT_HEADER, false);
curl_setopt($curl, CURLOPT_FOLLOWLOCATION, true);
curl_setopt($curl, CURLOPT_URL, $base);
curl_setopt($curl, CURLOPT_REFERER, $base);
curl_setopt($curl, CURLOPT_RETURNTRANSFER, TRUE);
$str = curl_exec($curl);
curl_close($curl);

// Create a DOM object
$html_base = new simple_html_dom();
// Load HTML from a string
$html_base->load($str);

//get all category links
foreach($html_base->find('a') as $element) {
    echo "<pre>";
    print_r( $element->href );
    echo "</pre>";
}

$html_base->clear(); 
unset($html_base);

It gets all the links as expected:

And make sure you have php_openssl and php_curl installed...