Melinda asked this question in a prior post.
The absolute best way for someone, vegan or not, to get more relative omega-3 (that is, to get a higher omega-3:omega-6 ratio) is to decrease omega-6 intake. You don't even have to take a pill.
You can take all the omega-3 you want, in pill or food, but if you're not also decreasing omega-6 you're not benefitting from it as much as you could. When omega-3 is high relative to omega-6, different compounds get made downstream, like the longer chain EPA, DHA - that's good. It's not the absolute amount of omega-3, it's the relative amount that matters.
Vegetable oils are high in omega-6. Limit nuts, seeds and oils made from them. Limit grain oils like corn oil, rice oil, and soybean oil. Do not add oil to food or eat food made with oil. The grain itself, like corn or rice, is fine because it's low in fat and so provides little omega-6 per serving.
Spreadsheet (This shows the ratios)
Above is a chart I made that shows omega-6 in some foods. Note that greens (romaine, broccoli, kale, spinach) generally have more omega-3 than omega-6 and so are excellent ways to improve your ratio. Eat them ad libitum, as much as you want. Walnuts on the other hand have more omega-6 than omega-3 and so are not a good way to improve your ratio. Although flax is high in omega-3, eating lots of it isn't a good idea. See below.
Both omega-3 and omega-6 are polyunsaturated fats. If you choose to take extra omega-3, in pill or food, instead of or in addition to decreasing omega-6, know that you are intaking more polyunsaturated fat. Fats that are not saturated have many highly-reactive double bonds and oxidize quickly, especially when they hit the acids in your stomach. Oxidized fats, and their metabolites, have been shown to increase risk for atherosclerosis and some cancers.