On Instagram, you can directly send messages to a particular person and chat with them. But if you are not able to send messages to the person and are facing the issue “Why can't I reply to messages on Instagram” then here are the things that you can try to solve this issue:
Update your Messaging feature on Instagram, for that go to the settings in your profile click the “Update Messaging” option, and then click on the update button. If you don’t find the option of “Update Messaging”, then you have to update the entire app instead.
Check your internet connection properly and connect your device to a stable internet connection.