Good vegan sources of iron include legumes (lentils, beans, and peas), dark leafy green vegetables such as spinach and kale, dried fruit, and blackstrap molasses.
A systematic review and meta-analysis of the effect of vegetarian diets on iron status in adults concluded that “vegetarians are more likely to have lower iron stores compared with non-vegetarians. However, since high iron stores are also a risk factor for certain non-communicable diseases, such as type 2 diabetes, it is recommended that not only vegetarians but also non-vegetarians should regularly control their iron status and improve their diet regarding the content and bioavailability of iron by consuming more plants and less meat.”
 
According to Jack Norris, Registered Dietitian and Executive Director of Vegan Outreach, you do not need to worry about iron if you are otherwise healthy and eat a varied vegetarian or vegan diet. If you think your iron stores might be low, you can increase iron absorption by:

• Adding a source of vitamin C at meals (see table of good vitamin C sources here). 
• Avoiding tea and coffee at meals.
• Increasing legume (peanuts, beans, lentils, peas) intake.
• Cooking foods (especially water based acidic foods like tomato sauce) in cast iron skillets.

Learn more at:

• Iron
• Iron Nutrition: Why the Rules are Different for Vegans