Eczema has three primary causes: (1) food allergies ; (2) allergies to cleaning products that you are using; and (3) stress (stress causes a build-up of cortisol, which irritates nerve endings.) You may want to be tested for food allergies to see if anything you are eating is causing the eczema. Vegans usually rely on foods that can be allergic, such as soy and nuts (allergies to nuts are related to tree allergies.) Also, since you are vegan, be sure that you are taking in at least 25-30 mcg of zinc each day by food or supplement. Zinc plays an important role in skin health, and vegans often don't have a sufficient source of zinc through food sources.
With respect to cleaning products, you may be using cleansers that you're allergic to, particularly cleansers with fragrances. The first time I got eczema was when I started using a soap that had tea tree oil in it, which I was allergic too. The eczema got all over my hands. Then it went away after I stopped using the soap once I figured out that I was allergic to it. The next time I got eczema was when I used a botannical shampoo that had something in it I was allergic to, and the eczema got all over my shoulders. If you are allergic to cleaning products, it's important to use only hypoallergenic, fragrance free products.
One of the best hypoallergenic soaps is Cetaphil gentle skin cleanser for sensitive skin. Don't get the general Cetaphil cleanser, though, because it has fragrances in it. Dove hypoallergenic soap for sensitive skin and Aveeno fragrance free boy wash are also good. Hypoallergenic, fragrance free shampoos are harder to find. While manufacturers are quick to make cleansers for sensitive skin, major manufacturers haven't seemed to come up with a shampoo for sensitive skin (or at least I haven't found one.) One I found is California Baby (for sensitive skin.) It is as hypoallergenic, and it cleans hair well. You can find the website by Googling "California Baby." Check your drug store or grocery to see if any of them carry others. I can also get away with Pantene Ice Shine, even though it is very slightly fragranced.
If you are allergic to fragrances, it's also important to stop using any detergents and soaps that have them because they will irritate your skin and may cause eczema. By using laundry detergents and fabric softeners that are fragrance, your skin won't be so irritated. I use fragrance free Tide and Downy, but there are others.
To make the eczema that you currently have get well, several on this board have had success with Doctor's Defense Zen Fusion Creme. It only helps to cure the eczema, though, so finding what is causing it still takes trial and error. One other suggestion is to start reading through these posts to learn from others what worked for them, and then start experimenting.