GrabzIt can attempt to screenshot mobile websites, but only if the target website has a mobile version of itself otherwise it will just display the standard website.
To create a screenshot of a website as seen by a mobile device using GrabzIt's API set the requestAs parameter to mobile and set the browserWidth parameter to be the width of a mobile browser. The recommended size is 320. To create a tablet screenshot don't set the requestAs parameter instead just set the desired browser width, a good size would be 768.
These parameters can also be set when using the Online Screenshot Tool.