Most hotels are willing to accommodate special requests, including providing a fridge in your room. You can reach out to the hotel directly by phone or email and inquire about their policies regarding fridges.
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Many hotels have a mini bar system that electronically records when you open the door to the mini fridge, when you pick up an item, and if you return it or not.
One way to keep food cold in a hotel room without a fridge is to use an ice chest or cooler. Fill the cooler with ice and place your food items inside. This will help keep them cool for several hours, depending on how much ice you have and how often you open the lid of the cooler.
Hotel staffs are usually very observant and aware of what is going on in the hotel. They will likely notice if a guest takes towels from their room, as it is not something that typically happens during a stay.
If that's the case for your travels, your best bet is to alert the front desk manager and ask to store your items in their kitchen refrigerator or ask that they set your room's mini fridge to the appropriate temperature. Or, if you can set it yourself, make sure the mini fridge is set to 40 Fahrenheit or below.