I'm not quite sure if there are any plugins that work out of the box the way you describe. You might want to do some research on the many importing plugins published for WordPress to see, if any of them support front-end importing with some kind of capabilities management.
I think that the most probable situation is that you need to do some custom development either by yourself or hire a developer to do it for you.
Based on your question and the related comments I think the concept for the import feature could be something along these lines.
- Create a custom form with all the necessary input fields and a file input field. You could perhaps have a custom shortcode render the form on the front-end.
- Create a custom user role with custom capabilities for those who need to be able to access and use the import form. With custom role/capabilities it might be easier to ensure the users won't be able to anything inappropriate. Which might be the case if you hand out admin or editor roles to users.
- Handle the import form posting in any suitable way. You can use PHP to extract data from a csv file, if there's one posted with the form. There's a good thread on How to extract data from csv file in PHP on StackOverflow.
- While / after extracting data from the csv file use wp_insert_post to create a custom post for each line of the file and add_post_meta to the posts as needed.
This is just a concept and I'm sure there are quite many details that need to be figured out and filled in before it works. But, this is the best answer I can give you based on the information you've provided thus far. I hope this at least gives you some new ideas on how tackle the situation and helps you come up with a proper working solution to match your needs.