I can't think of any reason why plants wouldn't be able to get cancer, but I've never heard of a plant growing a tumor. I've also never seen a plant with a noticeable abnormal growth. Can plants get cancer?
Yes, plants of all sizes can have cancerous growths.
Agrobacterium tumifaciens, the causative agent of crown gall disease, produces what is called a tumor. See this link for detailed information on these growths. Alternatively, use a plant physiology textbook to look up the above terms. (Here, is where a textbook is better than a single abstract in PubMed.)
I am certain that you have seen such growths on trees, such as this. Other smaller versions can occur on smaller plants - on stems and leaves, for example. The plant builds up tissue around the A. tumifaciens infection in an effort to isolate and contain the infection and its effects.