So, just to be clear, is this one of those cases where a deponent verb takes an ablative as an object? Or am I completely mangling the grammar here?
There is no ablative here. Hispanice is an adverb. That's the common way to put it when describing 'how' someone speaks (tacite - quietly, celeriter - quickly, etc.), or in what langauge.
Latine loquimur - We speak Latin.
It's an adverb. if you like, a literal translation could be: "The crowds speak Spanishly"