Is it permissible for me to access my friend's husband's bank account—without his knowledge—and inform her of its balance, to be used as evidence in her demand for alimony?

It is forbidden to look at a man's private account, as it is a form of spying, and Allah and His Messenger have prohibited it. Regarding the husband's stinginess: If he provides the necessary legal maintenance, his wife is not permitted to seek a divorce or spy on him, because seeking a divorce without a clear harm is not permissible. If he does not provide for her, she may refer her case to the Sharia courts to compel him to provide maintenance or to seek a divorce. If the stinginess is concerning luxuries, she must fear Allah and refrain from seeking a divorce, as the husband is not legally obligated to provide them.