There are a number of different ways you can centrally store information in Office 365 for contacts outside of your organization. Which option you should choose largely depends on how your users need to access and update the information.
Exchange Online Contacts – allows you to create external contacts that will appear as part of your Global Address List. Users can find the contacts in their Outlook Address Book or by searching the company directory on their mobile device. You can also add the contacts to distribution groups. However, only a user with Office 365 administrator privileges can create and update the contacts. In addition, each contact must have an email address listed. The contacts can be created one at time through the Office 365 or Exchange admin centers, but can be added in bulk utilizing Windows PowerShell.
Exchange Online Shared Mailbox – creates an additional mailbox in your Office 365 account. You can share just the contacts or give users access to the entire mailbox if there is other information you would like share, such as a group calendar. Given the correct permission, all users can create and modify the contact entries. They can access the shared mailbox information from Outlook or Outlook Web Access, but not via the contacts on their mobile device.
Exchange Online User Mailbox – creates a new user with an associated mailbox in your Office 365 account. As with a shared mailbox, users with the appropriate permission can view, create, and modify contacts via Outlook or Outlook Web Access. Although you have to add an additional Office 365 license to your account for this option, it does allow users to add the mailbox and its contacts to their mobile devices in addition to their own account.
SharePoint Contact List – allows you to store and display information for your contacts on your SharePoint Team Site. In addition to being able to customize the data you store for the contacts, you can create different views to help your users easily access information they frequently need. Through their browser, users with the right permissions can add and modify contacts, make bulk changes, and easily export information to programs like Excel. While you currently can connect and view your SharePoint contact list in Outlook, that functionality is slated to be phased out.
Of course if you are looking for a true Contact Relationship Management solution, you may want to look at adding Microsoft Dynamics 365 to your Office 365 account.