You are only able to send an Instagram DM (direct message) from within the Instagram app on your devices. Logging in to your account on a computer browser will allow you to check out profiles, pictures, followers, and stories, but you are not able to send an Instagram DM from the web. That could change in the near future
According to a recent tweet by app researcher Jane Wong, backed up by screenshots (below), Facebook is testing Instagram DM functionality for browsers. If the company decides to adopt the feature and roll it out to users, you will no longer have to rely on your mobile device to initiate or reply to direct messages.
Before you get too excited, just because the feature is being currently tested, it doesn’t mean it will make it (or soon) to the mainstream. However, it’s reassuring that Facebook is at least considering the possibility.