I don’t think the issue is caused by settings in a specific Cloudflare account.
Problem is occuring when the bot tries to access links from other domains, not just Zendesk.
Did some more investigation, I believe Cloudflare started to block requests coming from AWS EC2 IP addresses. When calling the links from my home laptop using curl or the bot, they work. When calling from an EC2 VM (tried different AWS accounts), I get the 403 error.
Anybody knows how I could get in touch with Cloudflare to whitelist my bot as a good bot? I filled out this form months ago but heard nothing back from them.