Does Azure Search support some way of ignoring accented characters? For example, if somebody searches for e it should include é characters
Yes. Please use the ASCII folding analyzer on your field. To do that, set the analyzer property on your field to
analyzer:"standardasciifolding.lucene"
Alternatively, use a language specific analyzer e.g. analyzer:"fr.microsoft". To learn more about analyzers in Azure Search take a look here.
Note: Different language analyzers treat diacritic marks differently, use the Analyze API to test analyzer behavior.