A simple approach is to make a contact form for prayer requests using a plugin like Contact Form 7 or Ninja Forms. You would receive prayer requests by email (Ninja Forms can save them into your WordPress admin area).
There are also several WordPress plugins for prayer requests that you may want to try.
We have not tested these yet so please let us know how they work for you.
You can search for others in the plugin directory or on Google: http://wordpress.org/plugins/