@Stuart Daly Welcome to Microsoft Q&A, Thank you for posting your query!
Firstly, apologies for the delay in responding here and any inconvenience this issue may have caused.
Yes, you can configure Azure Files to allow access to files via REST (HTTP/HTTPS) using SAS keys. Get File (REST API) - Azure Storage | Microsoft Learn
For web browsers they may run into a conditional headers issue and the workaround is here:
https://video2.skills-academy.com/en-us/azure/storage/files/storage-troubleshoot-windows-file-connection-problems#error-conditionheadersnotsupported-from-a-web-application-using-azure-files-from-browser
Hope this helps!
Kindly let us know if the above helps or you need further assistance on this issue.
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