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It is certainly feasible to do it on your own. You can get information about aurora conditions on the website of the Icelandic Met Office. It shows expected cloud coverage for the next few days and (except during the summer) a forecast for aurora activity.
If you are not a part of a tour, it is important to monitor this so you know when to stay up late! You'll need to make sure that you've got internet access.
You will also improve your odds of seeing northern lights by not staying near ReykjavΓk. Generally, your best odds of seeing them is to stay in the northern part (near Akureyri). There are never any guarantees though. During December, hotels will rarely be fully booked so you might leave it open and let the weather decide which part of the country you head for. That is a luxury that the tours will not afford.
Also, if you do decide to go on your own, keep track of weather reports not just for the northern lights, but also for driving conditions. December isn't usually that bad but there can be storms and road are occasionally closed.