The answer to this question is both yes, and no. For all intents and purposes, no you cannot play League of Legends without a graphics card. When you take into consideration that games need graphics in order to run, this is to be expected.
Even if you have a lower quality GPU this will still allow you to run League of Legends to an extent. Granted, it will not play well in comparison to when you are using the correct graphics card, but it will still be able to operate. Without a graphics card, the running of League of Legends will not be possible.
However, the answer is both yes, and no, because there is a way of getting around not having a graphics card. If you have an integrated GPU in your PC or laptop, then this will successfully run LoL. While the gameplay will not be as good as using a standard GPU, iGPUs are still a reasonably good alternative.