Dear David,
Good day!
Thank you for posting to Microsoft Community. We are glad to assist!
Based on your description regarding "How to delete a custom emoji in Teams?". Kindly note that Teams admins may limit who can delete custom emoji , Kindly check Use Custom Emoji in Microsoft Teams - Microsoft Support [2045-3261-d1e-92b6]

(If you are not sure who is your admin, can refer this How do I find my Microsoft 365 admin? - Microsoft Support).
I therefore believe that this issue should be further investigated by our online technical support team. A support Engineer can further connect with you remotely, for further troubleshooting and involve the backend team where need be. The team might request for your public IP address or collect certain detailed logs information with more technical resources. Please understand that as public forum support, we don’t have higher permissions to check and analyze these logs or check the IP addresses and considering your scenario maybe inconveniencing, to quickly help assist, kindly contact your organization admin or you can create a service request to report this symptom to our online channel. Here is contact support document: Get support - Microsoft 365 admin | Microsoft Learn
Important Note: If any organization's Office 365 Business / Enterprise subscription is from syndicate or reseller partner and if global admin cannot open service request from their side, they may need to contact reseller support provider so that they can help to global admin for opening service request from their side, after that Office 365 technical support team can involve in service request. Thanks for your precious time and your understanding would be highly appreciated. Have a nice day!
Best regards,
Inema |Community Moderator