This information seems to be available on the module level utilising the API. 

Reaction [R00352][1] points to the module [M00173][2]. While some information is in the `Definition`, it's not trivial to link it to the particular reaction. This linkage is however easier accessible via [API][3]: 

 [![Reaction orthology defined on module level][4]][4]

It seems to be a modelling issue that this additional lookup is required instead of this data being readily available on reaction itself.

  [1]: http://www.genome.jp/dbget-bin/www_bget?rn:R00352
  [2]: https://www.genome.jp/kegg-bin/show_module?M00173%20R00352
  [3]: http://rest.kegg.jp/get/md:M00173
  [4]: https://i.sstatic.net/orSvB.png