Yes, indeed, your client can publish items although the exact specifics of what they can and can't do depends on what level of access you’ve given them.
Basic users are unable to publish items, but are able to submit and view them. Please note, they cannot view items that they have not submitted to the account themselves.
If you would like your client to be able to publish items, it is best to upgrade them to either an Expense Approver or Admin User account.
If you do not wish to give them the access associated with an expense approver or admin account, but do not want to manually publish all of their items, the easiest solution is to use supplier rules.
Supplier rules allow you to automate the item submission process, setting default categories, payment due dates, payment method and more. You can also turn on auto-publish, which will automatically publish items from selected suppliers.
You don’t need to worry about your client interfering with the process, whilst simultaneously saving you the trouble of reviewing and exporting items from the suppliers you’ve set to auto-publish.