Is it possible to declare multiple file handles globally within an external module and use them on multiple function calls to that module?

Yes. There are several ways to do this. Or rather, there are several ways to hold open filehandles inside a module such that you don't have to open them in the calling scripts and pass them in, which is what I think that your really asking. Correct me if I am wrong.

The essence of all the different ways is to not use globals, or at least not named globals.

You can use a scalar, including a lexical (my'd one) in place of a GLOB as a filehandle.

open my $fh, '>', '/logs/mylogs' or die $!;
print $fh 'Stuff to end upin my log';
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See perlopentut for more information.

There are other methods, like creating an anonymous glob for each caller or using one of the standard modules like IO::File or IO::Handle.

I have a feeling that your question may be more to do with how to differenciate between multiple simultaneous callers, but that's a guess and we would need a clearer explanation of what problem, conceptual or actual, that you are having, before we would know how to properly respond.