Yes and no. It’s true that richest food in iron is red meat, but it is also found in some vegetables. By eating soy foods regularly you stand a good chance to get the same amount of iron as a person on a normal diet would. In 1999 a study was published on this topic in an American journal of nutrition, it has been demonstrated that there is no major difference between the amounts of iron ingested daily through a vegetarian diet compared to the iron intake of omnivores.
The problem of a vegetarian diet is rather a problem of storage. The body must store a large amount of iron for later use. Vegetarians store very little iron, while consuming as much. Iron from plants is absorbed and is metabolized more slowly than the one from meat. Some substances in vegetables, in large quantities, limit the absorption of iron. There are meat alternatives in vegan food industry such as vegan meats which are all plant based and rich in iron. Also your iron absorption levels are lowered by drinking caffeine beverages and by some spices too.