It's certainly possible; I'm not aware of any plans to implement this in the near term, but that doesn't mean it's never going to happen. If you have the inclination, I would encourage you to check out the Ideas Forum on the HubSpot Community. That's the best place for product feedback/ideas, as it's monitored closely by the product team.
Regarding the endpoint @pherris discovered; as he said, since that's not documented it's not officially supported, and is subject to be changed/deprecated at any time. I'd avoid building anything that heavily depends on that endpoint, in case there are any unannounced breaking changes.
Retrieving and storing meeting/call types against Engagements
Plus one for the need for call types against Engagements, I would need to filter on the call type to see how many of a certain kind we have (grouped by team or deal, but that can be fixed at a later state).
As of now we have to create a report, export it, loose all non latin letters, try to match by name to the keys in our global BI solution at the import.. well you get the problem)
It's not currently possible to pull a portal's call/meeting types. If you know the values, you can include them when creating an engagement, however. Using the example JSON from the Developer Docs & adding an {{activityType}} field: