I know that iceboxes were the only real way to keep food cold during the era before refrigerators, but the ice was usually delivered in trucks or at least carriages.
If the Venetians used iceboxes, was ice delivered by boat? Was it possible for them to use saltwater ice?
There's other ways to preserve food than ice.
You've got the good old salt, which has been used for millenia. It works well for pork, beef, fish, etc...
Also you can pickle food, with the acidity preventing the food from spoiling.
Even more simple, you can dry things, like fruits, to have them last longer.