Hello Sander Koster,
Welcome to the MS Q&A platform.
Unfortunately, Cookies (as output from HTTP Connector) are not supported with ADF, and PG is evaluating this as a feature request.
Here are a few feedback items submitted for this feature request.
https://feedback.azure.com/d365community/idea/b79ada56-7226-ec11-b6e6-000d3a4f032c
https://feedback.azure.com/d365community/idea/d4649a65-6e26-ec11-b6e6-000d3a4f032c
https://feedback.azure.com/d365community/idea/a01b3576-7126-ec11-b6e6-000d3a4f032c
All the feedback shared in this forum is actively monitored and reviewed by respective product owners. This would be open for the user community to upvote & comment on. This allows product teams to prioritize idea against our existing feature backlog effectively and gives insight into the potential impact of implementing the suggested feature
I would suggest leaving a comment and up-vote for this feature request.
I hope this helps. Please let me know if you have any further questions.