There are different scenarios where a user might encounter problems with editing custom fields. For example, when submitting a request, while viewing a custom form on an object, or when viewing custom fields on a report.
Step to resolve:
Ensure that the form which contains the field is attached to the object one is trying to edit.
If a non-admin user wants to attach a form to the object, they will need to manage access to the object in order to attach the form.
Verify all the details in the form sharing are correct.
Open the Sharing settings for the object to which the form is attached. See below:
Look for the affected user on the Sharingmenu.
If the user is not listed, add them to the Sharing menu.
Ensure that the user has the Contribute permission with the Edit Custom Form enabled in the Advanced Settings. See below:
(This step is optional) If this form is part of a Request Queue, check the sharing settings on the Request Queue custom forms, as well. See below: