Hi,
Based on my knowledge, whether users can access the mailbox, it depends on your Exchange Version.
After the migration is done, the emails starts to be delivered to Office 365 mailbox.
By the way, in this forum we mainly discuss questions and feedbacks about Office client products, since your question is more related to the migration to Office 365, please note:
1. For Exchange on-premise questions, please post the question in Exchange Server forum:
https://social.technet.microsoft.com/Forums/exchange/en-US/home?category=exchangeserver
2. For questions about Office 365, we have this Office 365 Community Forum:
http://community.office365.com/en-us/f/default.aspx
For further assistance upon this topic, you can post the question in the appropriate forum.
The reason why we recommend posting appropriately is you will get the most qualified pool of respondents, and other partners who read the forums regularly can either share their knowledge or learn from your interaction with us. Thank you for your understanding.
Regards,
Melon Chen
TechNet Community Support