Is it possible in SPFX to make a REST API call with a service account instead of the current users’ credentials?
You can call an external API from SPFx which is a recent possibility using the AAD authentication. Check out this link https://developer.microsoft.com/en-us/office/blogs/calling-external-apis-securely-from-sharepoint-framework/
But this is going to be an API call & anyone who have authenticated in to your AD will be able to call this API. You need to manage your security within your API layer.