The problem may arise from the Power BI Server's security settings, which do not allow anonymous web connections for scheduled refreshes. Power BI Report Server has much stricter authentication policies than its service counterpart, therefore requiring explicit credentials or gateway configuration.
Possible Solutions:
Use a Data Gateway: If the web source supports an alternative authentication mechanism (for example, API keys, basic auth, etc.), configure an on-premises data gateway to manage the refresh.
Change to a Different Mode of Authentication: Some web sources will allow switching from anon to basic authentication. Check if you can factor in credentials.
Workaround with Power Automate: Automate data extraction via Power Automate (formerly Flow) to cover the web data in a storage area accessible to the Power BI Server (e.g., SQL Server, SharePoint).
Do Manual Refresh in Power BI Desktop: If nothing automated works, consider doing a manual refresh in Power BI Desktop and republishing to the Power BI Server.