Several studies have shown the positive impact that vegan diet can have on gut microbiota1. At the same time several studies have shown the connections between a gut in a poor status and the occurrence of allergic reactions1,2. Can vegan diet play a role in the reduction (or elimination) of allergic symptoms? Has any study been conducted on this topic?
I assume you are asking about allergies unrelated to foods that are normally excluded from a vegan diet like eggs and dairy. So the question would be whether a vegan diet might reduce symptoms from allergies to non-food items like pollen, dust, mold, animal dander, etc.
An epidemiological study on diet, asthma, and allergies from 2001 showed that young adolescents (age 13-14) had decreased symptoms of wheeze and allergic rhinoconjunctivitis (runny nose, sneezing) associated with increased consumption of cereal and rice, protein from cereals and nuts, starch, as well as vegetables and vegetable nutrients. A more recent meta-analysis from 2017 concludes:
Fruit and vegetable consumption appears to be protective against asthma.
And a recent 2016 publication suggested that high-fibre diets may be beneficial for those suffering from allergies.
... gut microbiome-derived short-chain fatty acids produced from a high-fiber diet have been shown to protect against allergy.
I also came across non-scientific anecdotal evidence of allergy relief after adopting a vegan diet.
I went vegan around March/April last year, so right before summer. I can definitely say that last year my pollen allergy was a lot weaker than it had been the years before.
The evidence is far from conclusive, but it seems safe enough to recommend a diet higher in whole fruits and vegetables (and other foods containing dietary fibre) to somebody looking for ways to mitigate allergy symptoms.