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: