I think this is a great idea.
With the current implementation, this tool, which is the most convenient way to add images to posts, leads to a default behavior of not including an alt-text, which is an important element for readers who are visually impaired as well as for search-ability. Once the user has finished using the tool, the image is added, and the only prompt to add alt-text is some text in the post's markdown code that, most users will likely assume, will never be read by anyone. As a result, there is very little incentive and a little bit of extra friction to adding an alt-text, and therefore, most of the time, it doesn't happen.
This tool should include a text-entry box for alt-text, accompanied with some explanation along the lines of what is currently in the Markdown help for images:
The word in square brackets is the alt text, which gets displayed if the browser can't show the image. Be sure to include meaningful alt text for screen-reading software.
The explanation should link to more information about what should go in the alt text.
The alt-text field should be required, but there should be a checkbox labeled "Leave alt-text blank. This image contains no information," which, if checked, disables the alt-text field and makes it not-required.
While we're at it, it might make sense to also add an optional field for "Tooltip," so the user can do the entire image-entry process in one step, if desired.