You can store most things in cold water. If you are in a hotel room, use the bathroom sink, they are normally porcelain/stainless steel which is a cold material by default, put the plug in securely and fill it with a small amount, you don't need to fill the water to the top, it'll keep anything cool for several hours as long as it is not in direct sunlight, which they generally aren't. It also depends if you have something that is already open, use the drinking glass if they have provided one, instead of toothpaste and brushes put your vials in, or, whatever, you can stand it upright and only fill the sink with a very small amount of water, so, it won't tip over, otherwise, get something like a plastic food zipper bag that you put in your kitchen fridge, and put your vial and needles in that and place that in the sink instead, the bag will be cool and watertight for hours.
If you go away in the Winter months, you can use snow, if your hotel room is not in direct sunlight during the day you can pile snow up in one corner of your verandah, if you have one, and stick anything in it to keep it cold for hours, the snow won't melt quickly, if you can store bottled beer in it, or tin cans, you can store anything in there, be imaginative.