Lunapic fully supports image transparency and offers a few ways to add transparency to images.  

The quickest way if you have a plain background is to use Menu -> Edit -> Transparent

You can also use the scissors tool to cut out areas of the image. That's Menu -> Edit -> Crop.

LunaPic now support Alpha Transpency (translucence) Menu -> Adjust -> Alpha Transparency 

To preserve transparency, you need to save the image as gif or png.  

Example of image transparency: