To add vCenter vSphere client in the All Apps list and connect with SSO, you can use VMware Identity Manager (vIDM) to authenticate users and provide access to vRealize Suite components. However, as you mentioned, vCenter does not support vIDM as an identity source, so you cannot configure SSO for vCenter directly.

Here’s a possible solution:

1. Use vIDM to authenticate users and provide access to other vRealize Suite components, such as vRealize Automation, vRealize Log Insight, and vRealize Operations Manager.

2. For vCenter, you can use the VMware Identity Manager Connector for vCenter, which allows you to authenticate with vIDM using a web interface. This connector is available in the vRealize Automation App Marketplace.

3. Once you have installed and configured the VMware Identity Manager Connector for vCenter, you can add it as an app in your vRealize Suite catalog, along with the other vRealize Suite components.

4. When users log in to the vRealize Suite catalog using their vIDM credentials, they will be redirected to the VMware Identity Manager Connector for vCenter, where they can authenticate and access vCenter.

Note that this solution does not provide SSO for vCenter directly, but it allows you to use vIDM as an identity provider for other vRealize Suite components, and provides a way to authenticate to vCenter using the VMware Identity Manager Connector.

I hope this helps! Let me know if you have any further questions or concerns.