HI! Welcome. I don't have eczema or psoriasis, so I haven't really paid much attention to what effects it. My very superficial understanding is that there are very few verified links between veganism and eczema and psoriasis.
But in general, a healthy diet should help. and a diet that is considered anti-inflammatory and full of antioxidants should help too. Not based on any medical studies - just common sense.
It seems to me the best place to start is with a Whole Food Plant-Based Diet (WFPB). WFPB is not necessarily a part of the vegan lifestyle. However, it is
totally compatible with a
vegan diet. There are several versions of WFPB. they are all pretty similar. The similarities vastly outweigh any differences.
Maybe the most well known is the
Forks Over Knives diet. It got its own documentary. Doctors Campbell and Esselstyn got starring roles. The "brand" that may be trending highest right now is Dr. Gregar's "
How not to die" diet. there are at least 6 other brands, but I think one that deserves to be mentioned is the Nutritarian diet as described in Dr. Fuhrman's book
Eat to Live.
All of these books have a ton of supplementary material. (and it's all interchangeable). They have their own websites, youtube channels, cookbooks, etc.
You said you were looking for a place to start and I would recommend you start with any one of those books. Or maybe the documentary
Forks Over Knives.
BTW, Many of the members here eat a WFPB diet.
And a good resource that I just discovered by accident is the Vegan Junction