The InDesign application does not work on a phone or tablet.
Thank you Derek, I realize that. I'm looking to change from InDesign to another software program that has the features I listed.
I hope someone else has been through this issue as well, or there is a way to do what is needed through another Adobe app.
I'm not sure that there's a page layout program that has the features that you've listed. You'll always need someone to open the INDD document(s) and enter the edits and then publish the revised manual.
Have you researched doing something that's web based? If I'm following what you're looking for correctly, Wordpress (WordPress › Blog Tool, Publishing Platform, and CMS) will allow you to edit the sections in the manner you're looking for; however, there would be trade-offs: You'll have to host the "manual" (via internet or intranet). Users of the manual would always have to access it online via the internet or corresponding intranet. Making a print version (if necessary) would not be easy.
If you want to consider a web based solution, then Adobe Muse might work for you (it comes free as part of the Creative Cloud suit). If you know InDesign you'll find it very easy to create a site with Muse and you can enable parts of the site so that your customer can make changes.