How can anyone add to a SharePoint list but only some can edit all fields?

Question

My office has a list on our SharePoint for product requests. We were trying to make it so that the "+Add Item" link was available to all who want to request a map but only specific fields such as Name or Description would be editable on their
end. Once the item is in the system, the SharePoint admins in our office could update things like Status or Analyst. However, after talking to the individual in charge of our SharePoint permissions told us that we cannot set this basic ability to "everybody"
and instead need to create a special group with limited permissions. Unforunately we have high-turnover between our office and other locations and see this causing a problem in the near future. Is there a work around such as creating a web app or form that
will push the necessary data into the SharePoint list or maybe just send it via email to designated recipients in our office to process it?

Answer 1

I do not see any simple solutions here.  You have various options, each with different limitations and each will require some effort to implement.  Mix and match as best meets your needs:
Create a standard list as usual*. In the lists's Advanced settings, in the Item-Level Permissions group, select the options Read all items and Create items and edit items that were created by the user.  This approach lets users create their own items
and edit them, and they can still view other's, but they cannot edit other's.  Note: some list templates do not provide the Item-Level Permissions group of settings.
Create a custom permissions level: copy it from Contribute.  Remove all of the Edit and Delete list permissions.  This approach lets your users create new and view existing entries, but not delete or edit them.
Configure Incoming Email, and email-enable the target list.  You can configure the list to accept email according to the permissions that person has with respect to the list.  Thus, for example, a person with Contribute permission level would
be able to send email to the list but a person with Read permission level would not.
Create the list.  This will also create default New, Edit and View forms.  Customize the New form in SharePoint Designer 2013 so as not to display the fields you do not wish the user to edit.
Create new site fields and content types: create one content type that displays only a few of these new fields and the other that displays all of them.  Add these to the list.  Configure a custom content type as the default for the list that displays
only a few fields.  Users will see this content type when they create a new entry.  Edit the New list form to display only this content type.  Edit the View list form to display only this content type.
References
Configure custom permissions in SharePoint 2013
Configure incoming email for a SharePoint 2013 farm.
Create a custom list form using SharePoint Designer
Create or customize a site content type