Integration with Weebly?
The short answer is no. 

The long answer is that Weebly is a proprietary system - you would not be able to integrate CiviCRM. However, you could use clunky iFrames and there is work in progress to have some CiviCRM forms embedded on external sites. 

Some small organizations may use a proprietary system for their website and host civiCRM with wordpress or drupal on a sub domain - that could be an option too. 

What is your current version of CiviCRM and what is your current CMS as those are factors.