Why can't my client view order history on their site?

The most common reason for this issue is that the user account is shared. Shared user accounts are not able to view order history, in order to protect the privacy of users ordering through that account. This setting is configured in Site > Site Users > [specific user] > General Information > "User Account is Shared".
For more information about shared user accounts, please see the User Account is Shared section in Ch. 045 Adding/Editing Site Users.