Currently, there is no way to send a popup message to all users of an organization in one go without installing any app in their Teams client. To request this feature, please follow this link to the Microsoft Teams Community and raise a feature request:
Microsoft Teams Community Feature Request
Thanks,
Nivedipa
If the response is helpful, please click "Accept Answer" and upvote it. You can share your feedback via Microsoft Teams Developer Feedback link. Click here to escalate.