Stack Exchange is paying for a special Imgur uploader, which has a massive benefit over most other hosting sites because the images posted there never rot. For this reason, images in questions and answers should be uploaded via the special uploader.
Upload images to Imgur with the keyboard
I can't help you with your screenreader, but I do know how to upload images with only a keyboard.
There are two ways: from a URL and from your clipboard.
For either of these methods, you first need to open the image upload popup with ctrl g.
To upload from a URL, make sure you have the URL in your clipboard and press tab, then press enter, and then paste your URL from your clipboard with ctrl v. Lastly, hit enter to submit.
To upload from your clipboard, you will need to first have the actual image in your clipboard (if your image is on your computer, you can just copy it with ctrl c from the file explorer). Then you'll need to paste the image into the uploader with ctrl v, and submit it with enter.
If you're successful, you will have markdown like this in your post, although these two lines may not be adjacent:
[![enter image description here]]
(You should probably replace "enter image description here" with a description of the image.)
If you don't get it right, you can hit escape to exit the uploader popup, then start over by bringing the popup up again with ctrl g.
Note that your image will need to be under 2 Mebibytes (MiB). If your image fails to upload, an error will be displayed, but it's plain text, so hopefully you'll know it's there.
Upload at your own risk. If you upload the wrong image, it will still be accessible via a URL unless you contact Stack Exchange (via the "contact us" link), and someone there will contact Imgur to get it removed.
Host elsewhere and ask for help
I said that images should be posted via the special Imgur uploader, but this doesn't mean that you have to do it. If it's easier, you can always host the image elsewhere and link to it in your post. Then, somewhere in your post, ask for help uploading the image. (You could say something like "I'm blind and I can't find the uploader. Can someone upload this image for me?")
Then, someone (really anyone) can upload the image for you.
If you have any questions (or if anything is unclear), feel free to ask.