Here we suppose the customer's email domain named: contoso.com, and customer's requirements as below:

• Customer’s email system hosted under Gmail, and email domain named: contoso.com;
• Customer plan to migrate user mailboxes to Exchange Online by IMAP migration;
• During the migration, customer need the mail flow co-existence between Gmail and Exchange Online.

To implement this scenario we will walk through the following steps.

1. Provision Office 365 tenant: contoso.onmicrosoft.com

2. Prepare the mail flow between Gmail and Exchange online:

1) From Gmail side, make sure it could send mail to Exchange online with email domain: contoso.onmicrosoft.com;

2) Exchange online side, add customer domain: contoso.com, and from accept domain configuration, set it as “internal relay”,

3) And create special Outbound Connector for domain: contoso.com, and set Smarthost as the MX record of contoso.com.

3. Migrate user mailbox from Gmail to Exchange online, and configure mail flow co-existence:

1) After we launch IMAP migration, for example, we migrated user1 to Exchange online, it will be stamped with two smtp addresses:

SMTP: user1@contoso.com

smtp: user1@contoso.onmicrosoft.com

2) At the same time, we need to add user1@contoso.onmicrosoft.com as additional smtp address for user1 at Gmail side;

3) And set user1@contoso.onmicrosoft.com as the target address, which means message send to this user mailbox, it will be forward to Exchange online side, as domain: contoso.onmicrosoft.com is point to the Exchange Online;

4) When user1 (Exchange online) wants send mail to user2 (Gmail), as user2 is not existing at Exchange Online side, and domain: contoso.com set as “Internal Relay” under “Accept domain” configuration, so the message will delivery to Gmail through special outbound connector (smart host point to “MX record for contoso.com under Gmail” ).

Refer document:
Create an Organizational Forms Library in Exchange 2013
https://www.slipstick.com/exchange/exchange-2013/create-organizational-forms-library-exchange-2013/

Step by step instruction:
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Step 1: Create an Organizational forms library in Exchange online

    1. Connect Exchange Online with remote PowerShell

1) $LiveCred = Get-Credential
2) $Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://ps.outlook.com/powershell/ -Credential $LiveCred -Authentication Basic –AllowRedirection
3) Import-PSSession $Session

    2. Create Public Folder Mailbox

1) New-Mailbox -PublicFolder -Name PFHierarchy

    3. Create Organizational Forms Library public folder and set EformsLocaleID

1) New-PublicFolder -Path "\NON_IPM_SUBTREE\EFORMS REGISTRY" -Name "Organizational Forms Library"
2) Set-PublicFolder "\NON_IPM_SUBTREE\EFORMS REGISTRY\Organizational Forms Library" –EformsLocaleID EN-US

Step 2: Publish forms into Organizational forms library

    1. Assign permission to the Organizational forms library

1) Add-PublicFolderClientPermission -identity "\NON_IPM_SUBTREE\EFORMS REGISTRY\Organizational Forms Library" -user admin -AccessRights Owner

           Add-PublicFolderClientPermission
           https://technet.microsoft.com/en-us/library/bb124743(v=exchg.150).aspx

    2. Publish custom form as an FDM file via Outlook 2013

1) Check “Developer” tab from “Customize Ribbon”

2) Create one custom form

3) Publish to "Personal Forms Library”

4) Open "Forms Manager" dialog: File, Options, Advanced, Custom Forms, Manage Forms

        5) Select the form and click "Save as", save it as *.fdm file format 

6) Click "Set" button, and select "Organization Forms"

7) Click "Install", browse and select the saved *.fdm file, follow the wizard to complete the installation

8) Then Outlook client will see this form from the Organizational Forms