Allowlisting domains is a very common task for IT security personnel, but the procedures to do so can vary wildly based on software, hardware, and cloud services.
The generic guidance would be to make sure that all internal firewalls and content filters allow "bit.ly" and "bitly.com" domains. Frequently we will see firewalls that block the entire ".ly" domain impacting the availability of Bitly links, so this would be a common use case to ask your security experts to check. The use of email-as-a-service companies like Google and Office365 can also complicate things. A Google search for "allowlist a domain in Office365" or a similar query can take you to the step-by-step instructions you need.