Kind of. Firstly plants make their own food from sunlight, air and water. The only way that a plant could not get enough food is if you kept it in a dark room or cut all its leaves off. Under those circumstances where plants are lacking in food, usually when they drop their leaves in winter, they utilise starch reserves. Those starch reserves caan be in the stem or they can be in the roots. In plants like carrots and sweet potatoes those starchy roots are quite obvious and are also a food source for humans. So in one sense you could say that some plants “eat their own roots”, insofar as a carrot root will shrivel up as it consumes its own starch reserves.