I have a form with a field for uploading images. When I am logged as an admin the image is uploaded, however, when I am logged as another user (with another role), I receive the following error: There was an error uploading your file.
I have set up Access to manage post types. The role in question has also "publish" capability within Media post type, and publish capability of the post type, where the form is included.
That will create a file debug.log in your wp-content/ directory which you can examine in any text editor. Try using the form again and then inspect the log. If you don't find the debug.log file it means it didn't generate any warnings or errors.
Otherwise, can you try disabling all non-Toolset plugins to see if the image upload works. If so you can re-activate your other plugins and try to identify which one causes the problem.
You can try the same with your theme, switching to the twentyseventeen theme and re-testing.