A good way, 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 amounts of compounds get made downstream, like the longer chain EPA and DHA.
Vegetable oils are high in omega-6, as are the nuts and seeds they are made from. Oils from grains like corn, rice, and soybean are also high in omega-6. Not adding oil to food is a great way to reduce omega-6. Whole grains, however, like corn or rice are low in fat and so provide 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. 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.