Unfortunately at this time it’s not currently possible to request a verified badge from Instagram. Only SOME public figures, celebrities and brands have verified badges. With this said, there are other ways to let other Instagram users know your authentic like:
You can link your official website, Facebook page or Twitter account to your Instagram profile
Posting videos and content of yourself that is not available anywhere else
I have a verified badge on Facebook… why don’t I have one on Instagram?
Even if you have a verified page on Facebook it does not mean your Instagram page will be verified. Instagram stats that at this time, only certain Instagram accounts that have a high likelihood of being impersonated have verified badges.