The default for mobile browsers is the mobile site theme, which has minimal design, and without the rich editor, meaning no file upload indeed.
However, the footer contains a link to the full site theme:
Tap the "full site" link and you'll see the full site on your mobile device. Then you can upload images from your device by tapping the Image icon:
As the dialog says, you can tap it (anywhere) to open the device file selection, which is different for every device/OS but you'll be able to choose the desired picture and upload it.
The only catch is the 2 MB limit, which is quite low for most modern devices with HD cameras, so you'll have to shrink the image first if it's bigger. Quick Google search brought this app for iOS (both iPhone and iPad) and this one for devices running android OS.