At this time, it is not possible to access campaign objects via the API. Could you describe your use case in more detail, though? How do you plan to use the campaign IDs?
HubSpot support may have covered this, but each HubSpot campaign's ID is visible in the URL. For example, in the URL:
the campaign ID is
e5d960a2-8386-4884-a021-ac84d81eacd9. This is one way you could get campaign IDs manually.
If you would like a public API for campaigns published, please upvote and comment on this HubSpot Community Idea. This information is sent directly to our product teams and helps prioritize feature requests that have the greatest impact on our customers. I’ve already upvoted the idea and will also do my part and pass your feedback along through internal channels.
Finally, could you share your support ticket number? I'd like to close the feedback loop for the team.