This is a summary of how these deployments are normally structured, not legal advice. Your DPO or counsel decides what is lawful for your business.
The single most important distinction. Almost all the relationship signal lives in the headers: who wrote to whom, when, and whether there was a reply. The body of the message adds very little and carries almost all of the privacy risk.
A vendor that reads only metadata is doing something materially different from one that parses message bodies, and it is a far easier conversation with your security team. Ask the question directly, and ask for it in the contract.
A lawful basis. Legitimate interests is the usual one for B2B relationship data. It requires a documented balancing test - your interest in the processing against the rights of the people whose data it is. Write it down before you deploy, not after.
Transparency. Both sides. Your employees need to know their mailbox metadata is being processed and why. The external contacts in the graph have rights too, which in practice means your privacy notice has to cover them.
Data minimisation. Collect the fields the product actually needs. "We ingest everything in case it is useful later" is the posture that fails an audit.
Deletion and access. You must be able to find and remove an individual on request. Ask the vendor to demonstrate this on a real record during the evaluation, rather than describing it.
Retention. Set a limit. Relationship data that decays in usefulness after two years should not be kept for seven.
In practice, failures are organisational rather than technical:
Orbb processes relationship data inside the customer's own tenant with their consent, works from interaction metadata and public professional history rather than message content, and supports per-individual removal. If you are running a security review, ask us for the detail rather than taking a web page's word for it.