This is literally what we've all been waiting for!
Instagram's just released a fully-functional web app that lets you upload photos from your mobile browser just like you would on the app and users have figured out a little hack that lets you use the mobile web app on your desktop.
Here's how to do it:
Go to Safari > Preferences > advanced. You'll then need to check the box at the bottom that says "Develop menu in menu bar".
Open up a private browsing window and go to Develop > User Agent > Safari -- iOS 10 -- iPhone. Then open up and login to Instagram which should now look like the app.
First login to Instagram on your computer. In the top right had corner, click on the on the three dots to open the drop down menu. Go to More tools > Developer tools. You'll see whole bunch of code open up in a sidebar on the right but just ignore all of that. All you need to do is go to the top left of the sidebar and click on the symbol that looks like a tablet and phone.
From here the page will change to look like how it does on the app so just click the camera at the bottom to upload your picture!