There are a couple of ways to send email without the need of your own SMTP server. There is an addition for PHP you can get called PHPMailer which will simplify the process. Or, you can send it with php's built in mail() function, but you must be sure to to format the header information correctly.
For PHP, this question has been asked before. See this link for details.
However, sending mail without using a verified SMTP server will most likely result in your email being sent to customers junk mail folder. If any emails are destined to corporate addresses, the corporations firewall may drop the email entirely. Typically if an email provider detects enough emails of this kind, it may even blacklist your IP / server for a time.