I would avoid wearing ankle supports unless you have to- can cause more problems in the long term as it trains your ankles to believe that they don't have to do the work for you and ironically can make them weaker.
It could be your footwear. A decent trainer shop should be able to offer you an analysis of your gait and running style and recommend the right trainers for you.
Alternatively, slightly collapsed foot arches (flat feet) can cause sore ankles after exercise. This is quite common and easily treated, but would need diagnosing properly.
You may just have weak ankles and there are some good physio exercises to help this (again check with your GP first).