There are 2 things to consider while answering this question: How to stop the radiation, and how to make the radiation.
There are 3 ways to "stop" radiation:
1) Time - Not an option, because... well it just doesn't make sense. Typically, time is considered a method to "stop" or lower radiation because the radioactive materials decay - that would mean our plant is dead.
2) Distance - Not an option, because if we put the plant brings the radiation with it.
3) Shielding - This is our option. It's possible for our plant to use a form of biological shielding. By growing extra "skin" around itself (fairly thick skin), it would be protected by radiation by the absorbent layer outside. This layer can be shed and regrown as needed such that the radioactive layer is never saturated.
The basis of your plant at this point: Thick stalks, thick leaves, thick everything.
But that still leaves us with how it produces radiation.
It just so happens that all things produce radiation, even humans. Living things require potassium, and in every 8,550 potassium atoms is one radioactive potassium-40 atom. Lets use Bananas for a second - they contain high amounts of potassium, and about 600~ bananas contains enough potassium to emit 1 chest x-ray of radiation. In a world where the plants are able to concentrate these nutrients (perhaps some radioactive handwavium is in the ground that it absorbs and moves) into certain "flowers" or "fruits", you now have radioactive appendages that poison the surrounding area, while the plant itself is protected from radiation via its thick "skin".
Your plant now possibly looks like a really thick, green glowing flower with fruits hanging from it's stalks. Perhaps it has really really big disposable flower petals that are radioactive, and drop off as it grows more, such that it can increase the radioactivity of the area surrounding it quickly when the wind carries the petals away and scatters it into the wind.