Your code syntax above is incorrect - the line:

open(FLATFILE, ">> /tmp/flatfile.data" or die $!...);
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..should read:

open(FLATFILE, ">> /tmp/flatfile.data") or die $!;
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I don't know whether this is just an error you introduced while posting this question otherwise it should have given you an error.

Apart from that I can't see anything wrong with the file writing logic which leads me to a couple of conclusions:

Are you sure the permissions are correct? Chances are when you run it via the form it's not using same user account that it does at the command line - and that account may not have permission to write (which is likely as most "web-user" accounts on servers have very few permissions by default for security reasons).

What platform/OS etc is this running on may I ask?