Yes the Twitter API supports JSON and technically it supports JSONP but you should not use it since it would compromise your application secret.
The easiest way to get started with PHP is TwitterOAuth. Example code to get tweets posted by a specific account:
$connection = new TwitterOAuth(CONSUMER_KEY, CONSUMER_SECRET, $access_token, $access_token_secret);
$statuses = $connection->get("statuses/user_timeline", array("screen_name" => "twitterapi"));