Not really.
Webstorage is more about string type of datas.
You can use it to store the progression in a game, store values about the player that you will to keep consistantly between two times he might launch the game/close the tab/open it again, etc...
In my game Drop Da Piew I used it to store the levels/scores/configuration options...
A PNG 8-bit compression on most textures will keep alpha channels (transparency), keep the quality and still majorly reduce the size of the images.
And if their original size is adapted to their display size you won't use more size than necessary.