I figured out how to use Curl to do it. My own login and password and merchant group of course are hidden in this code. But here's how to do it: I hope this helps anyone in the same situation.
$cookie_jar = tempnam('/tmp','cookie');
// log in
$c = curl_init('https://dashboard.powerreviews.com/dashboard/login');
curl_setopt($c, CURLOPT_SSL_VERIFYPEER, false);
curl_setopt($c, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($c, CURLOPT_COOKIEJAR, $cookie_jar);
curl_setopt($c, CURLOPT_FOLLOWLOCATION, false);
$page = curl_exec($c);
curl_close($c);
$post_fields = array("j_username" => "USERNAME", "j_password" => "PASSWORD", "command"=>"Login");
$post_data = http_build_query($post_fields);
$c = curl_init('https://dashboard.powerreviews.com/process_login');
curl_setopt($c, CURLOPT_SSL_VERIFYPEER, false);
curl_setopt($c, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($c, CURLOPT_COOKIEFILE, $cookie_jar);
curl_setopt($c, CURLOPT_COOKIEJAR, $cookie_jar);
curl_setopt($c, CURLOPT_POST, true);
curl_setopt($c, CURLOPT_FOLLOWLOCATION, true);
curl_setopt($c, CURLOPT_POSTFIELDS, $post_data);
$page = curl_exec($c);
curl_close($c);
// switch to Reviews Account
$c = curl_init('https://dashboard.powerreviews.com/programs/epc?accessMerchantGroup=xxxxx');
curl_setopt($c, CURLOPT_SSL_VERIFYPEER, false);
curl_setopt($c, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($c, CURLOPT_COOKIEFILE, $cookie_jar);
curl_setopt($c, CURLOPT_COOKIEJAR, $cookie_jar);
$csv_file = curl_exec($c);
curl_close($c);
// download RA unsubscribes
$c = curl_init('https://dashboard.powerreviews.com/dashboard/unsubscribes.csv');
curl_setopt($c, CURLOPT_SSL_VERIFYPEER, false);
curl_setopt($c, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($c, CURLOPT_COOKIEFILE, $cookie_jar);
curl_setopt($c, CURLOPT_COOKIEJAR, $cookie_jar);
$csv_file = curl_exec($c);
curl_close($c);
echo $csv_file;