We\'ve noticed that from time to time we will get a HTTP request without a valid User-Agent string. Is there any valid real-world case for accepting this type of HTTP request?>
From my experience, there is not Legitimate use case, at least no common one.
From my server logs, all requests without a user agent are malicious. All come from bots.
In theory, someone who "really cares about privacy" could do this. It depends on what service you're providing. I run a couple of websites. People not using browsers are not my clients.
While it's true that it's trivial for a developer of malicious bots to add a user agent (fake one like Chrome), it doesn't mean it's not illegitimate in intention.