Allowlisting domains is a very common task for IT security personnel. The procedures to do so can vary based on software, hardware, and cloud services used by your organization.
Our general guidance is to make sure that all internal firewalls and content filters allow the "bit.ly" and "bitly.com" domains. We sometimes see firewalls that block the entire ".ly" top-level domain. This impacts the availability of Bitly links, so ask your security experts to check.
The use of email-as-a-service products like Gmail and Microsoft 365 (previously known as Office 365 or Office Online) can also complicate things. A Google search for "allowlist a domain in Microsoft 365" or a similar query can take you to the step-by-step instructions you need.