What happens when you submit the contact form?
If you get the thank you message, then the mail should have been sent successfully using the built-in WordPress sendmail function.
Are you sure you put in real email addresses for the members in the member directory? And sent the test message to a member you entered with an email address you can check? Did you check your spam/junk mail folder?
This form does NOT send a copy to your wordpress admin email, so you won’t get copies of any emails sent to other members you have set up. If you want someone to use it to contact you, you need to enter yourself as a member in the member directory first with correct email address you want to get email sent to.
If you want to save copies of all emails sent through your contact form, you can install the Contact Form to DB plugin and set the options in my plugin to send data to it:
http://ww.wp.xz.cn/plugins/contact-form-7-to-database-extension/
(you don’t need to, and shouldn’t, install contact form 7… not required for the contact form to DB extension to work with my plugin, and may conflict with my plugin).
My plugin uses the built-in WordPress sendmail function to send the emails. So, if you didn’t get an error message, then WordPress told my plugin that it sent the email successfully.
If you are not getting the emails, that probably means that other WordPress emails are not getting sent either, and you have a configuration problem with WordPress or your server.
I recommend you install this plugin:
http://ww.wp.xz.cn/plugins/configure-smtp/
Use that to configure and test the mailing system for your blog. It has a nice test button that you can use to test the sending of emails.