Sure, long before we had electricity and refrigerators people on farms kept their food in spring houses, which were little storage buildings built over running creeks or natural springs. In fact the 180 year old farm my great-great-great grandpa built that my cousins still live on still has a spring house on the property and they keep beer in there during family reunions. There are probably instructions on line for building a spring house -- Mother Earth News might have something on their site.
check the temp of the creek. if it's 41 degrees or below it will work fine. you can dig a little hole in the creek bed out of the main flow of the water and make a sturdy box to set in the hole to put your food in. you don't want to put it where it would wash away. put the food in water tight containers. you don't want the creek water getting into any of the food, as there is bacteria and debri in the water.
We often use creeks to cool food and beverages when fighting fires in the mountains... plunk the canteen or plastic package right in the water, with stones or whatnot to keep them from floating away. Just make sure you rinse them off with potable water before opening things, to flush the bacteria away.
I would first take the temperature of the creek... if it's between 35 and 38 degrees F (1.7 to 3.3 degrees C) it would be perfect for a fridge
If it is colder than this the only harm would be to fresh foods like fruits and vegetables. Any frost to those could damage the delicate tissues...leaving them mushy.
Any warmer will NOT be advised....any increase in temp with significantly boost bacterial growth
http://www.alken-murray.com/Growth3x.jpg
If you wanted foods to last a day it shouldn't hurt...but if you wanted it to last a week at higher than fridge temps it will likely not work
I live on a piece of land with no house on it. There is a creek. I got out to eat all the time because I can't preserve food or have a kitchen except a grill. I am going to buy a propane stove and dutch oven, but I also need to preserve food. Since creeks are always really cold, could I just keep food in the creek?
