The purpose of this post is to recap the current studies on supplements that may help with Autism. With autism there are DNA mutations (SNP) that play a role as well as multiple different bacteria shift patterns.
Given that GI issues seem to be overrepresented in ASD populations, and that GI issues have been associated with a number behavioral and neurological deficits, dietary manipulation may offer a cheap and easily implemented approach to improve the lives of those with ASD.Dietary Approaches to the Management of Autism Spectrum Disorders. (2020)
“Vitamin B6 is beneficial for about half of autistic individuals in decreasing behavioral problems.”  and for this we can identify the ones that are more likely to react. ‘Statistically significant variables for vitamin B6 responsiveness, including combination of hypersensitivity to sound and clumsiness, and plasma glutamine level, were included.’ 
Neurological-Audio: hypersensitivity to noise has some a strong pattern for bacteria shifts as is shown here. It is likely that the vitamin B6 is altering several of these bacteria , hence the improvement (know to alter several hundred taxa).
Despite a considerable interest in dietary interventions, no consensus exists regarding optimal nutritional therapy. Thus, patients and physicians are left to choose from a myriad of dietary protocols. This review, summarizes the state of the current clinical and experimental literature on nutritional interventions for ASD, including gluten-free and casein-free, ketogenic, and specific carbohydrate diets, as well as probiotics, polyunsaturated fatty acids, and dietary supplements (vitamins A, C, B6, and B12; magnesium and folate).Nutritional interventions for autism spectrum disorder. (2019)
Dietary supplementation (including omega-3, vitamin supplementation, and/or other supplementation), omega-3 supplementation, and vitamin supplementation were more efficacious than the placebo at improving several symptoms, functions, and clinical domains.Dietary Interventions for Autism Spectrum Disorder: A Meta-analysis. 2019 (Full text)
- Omega-3 supplementation was more effective than placebo in treating the following symptoms and/or functions groups: language (general) and social-autistic
- Omega-3 supplementation was more effective than placebo in treating the following clinical domains: core symptoms and associated symptoms
- No details of which vitamins 🙁
- Omega-3 LCPUFA(722mg) with and without vitamin D (2000 IU/day) may improve some core symptoms of ASD but no definitive conclusions can be made. 
- Seventy-five percent of children with ASD consumed supplements with multivitamins (77.8%), vitamin D (44.9%), omega 3 (42.5%), probiotics (36.5%), and magnesium (28.1%) as the most prevalent. 
- Prebiotics only improved certain GI symptoms; however, when combined with an exclusion diet (gluten and casein free) showed a significant reduction in anti-sociability scores. 
- The two available double-blind, randomized, placebo-controlled probiotics trials found no significant difference in GI symptoms and behavior. 
- Effects of Lactobacillus plantarum PS128 on Children with Autism Spectrum Disorder in Taiwan: A Randomized, Double-Blind, Placebo-Controlled Trial.  – Positive impact
The research put forward that in individuals with ASD, while gluten-free/casein-free and ketogenic diets, camel milk, curcumin, probiotics, and fermentable foods can play a role in alleviating ASD symptoms, consumption of sugar, additives, pesticides, genetically modified organisms, inorganic processed foods, and hard-to-digest starches may aggravate symptoms.Current nutritional approaches in managing autism spectrum disorder: A review. 
18 randomized controlled trials of five supplements were included. B6/Mg was not helpful for improving ASD symptoms (seven RCTs). Two RCTs of methyl B12 reported some improvement in ASD severity but the effects on the correction of deficiencies were inconclusive. Two RCTs of vitamin D3 both reported increased levels of mean 25(OH)D in serum but inconsistent results in behavioral outcomes. Omega-3 fatty acid supplementation did not affect ASD behaviors but may correct deficiencies (six RCTs). One RCT of folinic acid reported positive results in improving ASD symptoms measured by various behavioral scales.Supplement intervention associated with nutritional deficiencies in autism spectrum disorders: a systematic review. 
Inconsistency of results would be consistent with microbiome variations. Working from the specific microbiome of a child or adult will likely have a higher success rate than randomly trying items that helps some subsets.