No, your clients will not have access to Task Manager. Task Manager is a project management tool built for fulfillment agents and other internal teams. You can learn more about creating Task Manager users for your team here. If you were to add a client as a Task Manager user, they would be able to see all accounts, not just their own. They would also be able to see private notes and adjust global task settings.
Although your clients cannot access Task Manager, it is possible for them to see updates on their fulfillment projects in Business App. Learn how to make fulfillment projects visible in Business App here.