Disclaimer: This could be all wrong.
So your computer only actually communicates with the game when you see the "saving" symbol in the corner. It saves at the beginning of every match you start to save the loss and lower score/electrum, and then it saves again after the match (either the same lower score/electrum, or with the increased win total and higher score/electrum). It also saves upon exiting the deckbuilder, bazaar, or card upgrading spot (I believe). You could actually not be connected to the internet, play for hours, and then when your internet reconnects, save then. You can also sell a card in the bazaar, change your mind, and close the game, since it doesn't save until you leave the bazaar.
For this to work, you would basically have to operate every match like a PVP match. If you've ever played a PVP match with someone in the same room as you, you know that it can take upwards of 20 seconds for the other player to register that you ended your turn and theirs has begun. You would basically be doing this every match against the computer, if you wanted it to save your progress. If you wanted it to save the progress locally on the computer, that might be possible (I don't really know), but none of the important data in this game is saved on your computer. If you had that local save, and then went to another computer and sold the cards in your deck, and then came back to your computer, what would happen when you loaded up that mid-game save? I don't think it will work locally, and for it to be done server-side would make the game unbelievably slow.
I could be wrong though. This is my interpretation of how the game works.