I've finally got some specific training booked for myself in a couple of weeks, but it's been 4 years coming! In this case, I informally discussed the need for, and the benefits of, such training and it met with agreement. But nothing happened, so in the end, I made a formal (PRINCE2) business case highlighted the benefits of this training and the risks with not training. It embarrassed them into action, but even so, it took 18 months to be realised. (politics!)
I'd recommend producing a formal business case, using whatever procedures you use in-house. If nothing else, it will distinguish your needs from the usual whining that we are all prone to. If your bosses don't believe in formal training you probably wont get anywhere, but at least you will know.
Note: in many cases, there is nothing wrong with learning on the job!
Anyway, make a formal pitch and you will more likely get a formal answer, and you may well impress them a little with your mature approach into the bargain.