The biggest challenge to a vegan is meeting the need for nutrition related to meat or dairy sources. If there is no meat in the diet, then Essential Fatty Acid (EFA) needs to be provided as the name suggests. In fact, it might be easy to not meet this need even on a non-vegetarian diet. The hard to get EFAs are the Omega-3 fatty acids which come from oily fish, and a few nuts that most people do not eat daily—pumpkin seeds, sunflower seeds and walnuts. Green leafy vegetables and corn are a source of trace amounts, although not concentrated. Shellfish is an option for EFAs, if you consider it one.
Otherwise, you will get a small amount of your EFA daily minimum from soy beans, or hemp oil, flax seed, and canola oil used in cooking. Ordinary peanut butter is a good source of EFAs. However, if a reliable source of Omega-3 (such as walnuts) is not plentiful in the diet, then it might be a good idea to supplement Omega-3 fatty acids in particular. While DHA and EPA, which are also from the fatty acid family, may not be depleted in a healthy person, some people may have a specific need for more.
Related to the Essential Fatty Acid story would be the Omega-6 oils, which are common in a typically oily diet. How the balance is achieved has yet to be settled, but the advice is a “recommended” daily allowance of no less than 1 gram of Omega-3 fatty acids for a female (5-8 walnut halves,) or about twice that for males, maximum advised is three grams.