It will work just fine. PHP runs inside of webservers that handle the work of multiple simultaneous requests. SQLite is also capable of handling multiple simultaneous requests, albeit not as many as a propers SQL server like MySQL.
EDIT
I just checked... The one-file version of PHP-login does have a primary key.
$sql = 'CREATE TABLE IF NOT EXISTS `users` (
`user_id` INTEGER PRIMARY KEY,
`user_name` varchar(64),
`user_password_hash` varchar(255),
`user_email` varchar(64));
CREATE UNIQUE INDEX `user_name_UNIQUE` ON `users` (`user_name` ASC);
CREATE UNIQUE INDEX `user_email_UNIQUE` ON `users` (`user_email` ASC);
';