In most cases there will be no disruptions. The only thing that might be interrupted would be if you are using DNS service with the current domain registrar. It's possible they will terminate that service upon transfer of the domain, so you might want to switch the DNS service either to us or a third party before you transfer the domain to us.
Assuming that your DNS is valid and continues to operate, there should be no interruptions in any services tied to that domain during the transfer.