A search through #plugins didn’t show many results for forms directly.
In your category settings, you can create “Topic Templates” which will autofill new posts made in that category with pre-determined text. A good example of this is to draft a topic in support and watch text appear in the box.
You might could create the form in that template field, then have form responses be made under that specific category, to which users can fill in the blanks and submit.
The only issue with this is that responses will be public. You can unlist posts, which will make them only viewable to the creator, admins, and anyone with the link. I did a quick search and @theblazehen was able to make a cron job that automatically unlists posts in a specific category:
You could remove the and closed = TRUE part to make it unlist all posts in the category. Make sure to change the category ID as yours won’t be 19.
Someone probably has a better suggestion, or even a full-on plugin for this, but I can’t think of any other suggestions that aren’t external.