You could always use a cooler, they are insulated. I don't know what brine you are making, but my brine only takes about 20 minutes to throw together. From adding the salt to simmering water to adding the turkey at the end. It is suggested that if you brine you do so for no less than 4hrs up to 24hrs.
Just so you know, I smoked a turkey like I do every year for Thanksgiving. I was going to brine it, but ran out of time. So I stuffed the cavity of the bird with 3 apples (any kind you want to use) and 3 oranges, all of which I quartered. I did however use an injection. Which was an easy one.
2 cups of a good chicken stock
4 tablespoons of unsalted butter (sweet butter)
3 tablespoons soy sauce
1 teaspoon poultry seasoning
just place all in a sauce pan, stir constantly on simmer until the butter melts, remove from heat and inject.
But yes I did not brine this year and the turkey actually turned out juicer than the years that I spent brinning.
Here is a picture , a before and an after.
Hope all this typing helps you out, and if you have any other questions just ask. :P