For the user profiles, we do not wish to display an email address for each user, but it is required. Also, we cannot simply put "Upon Request" either. Is there a change I can make in a child theme to allow for "Upon Request" to be put in the email field for the users. Or if possible could I make a change allowing the email field to be left blank.
Attached files: user field.PNG
User emails for user profiles is a WordPress default, and not a conditional manage by themes. I'm unaware of a customization that you can do to override this, and would recommend against customizing default/required user fields in WP since these are likely related to account recovery, and security.
That said, depending on your end goal, you may be able to do something to meet your endgoal, but it's passed the scope of support that is include with the theme. That said, based on what I know, I would approach it this way:
Add the correct email address to WP
Then you need to decide if this should be hidden for all, or just some. Note, the level of effort and the risk that is associated with each option may vary significantly depending on your business requirements, so it may make sense to go back to the business and provide them with what is technically feasible and what is recommended long run...
I'd suggest hiding all in the markup of your child theme, then providing it upon request, and linking your Upon Request with a Contact Us page. Alternatively, you may be able to mask these with a function, or with some js, then reveal based on some criteria... Again, and unfortunatly, I can't offer more than this since its beyond the scope of support provided here.