The ability for agents to modify end-user profiles changes according to your account's pricing plan.
The ability for agents to modify end-user profiles in these plans is defined by the access those agents have to tickets. However, agents are unable to modify organization fields.
To allow agents to change user fields for end-users, they need to have access to All tickets. Adjust the access an agent has to tickets in their user profile:
For more information, see the article: Adding agents and administrators.
If an agent cannot adjust the organization of a user, it is likely that the agent's custom role settings does not permit them to do so. For more instructions on how to adjust an agent's role, see the section: Assigning custom roles to agents.
Under the custom role page, the options to Manage user fields and Manage organization fields must be checked. For more information, see the section: Creating custom agent roles.