I have followed your guidance, and it's now possible for my authors to approve/publish using a Toolset form. However there is a big issue.
I have created a view for authors to use when managing theire posts. The view includes both published posts and posts that are Pending Review. In the wiew output I have included links for edit post and delete post.
For some reasons edit post links do not display for posts in pending review. Delete post links displays fine. Se attached screenshot.
I checked and it seems logical that if the post is not published, Toolset Form will not allow you to edit it from the frontend.
I do not see any way to Edit the post having pending review status from the frontend and its not logical as well.
Update: But I have a point, it shows post delete link not post edit link. Strange. I need to check this thing on fresh installation. I'll get back to you as soon as I know more.
Please do. I would like to request that this should be changed/fixed. If website visitors are submitting content that I don't want published without a revision, and I don't want the people managing content to have to work with the WordPress back end, I have no proper way to accomplish this. I can display a list of posts of any post status so not being able to edit them with Toolset forms is not logic at all, sice I can delete them using the same approach as you have seen.
Then, I re-check on your install and I found that the post-type "Gjestebok" was not set to manage by Access. So, I set that post type to managed by access as well as grant the permission "Edit Any" to the Author role.
=> hidden link
Please check the screenshot: hidden link
I can see it's working on frontend as well: hidden link
Can you please confirm it works at your end as well.