Last time I used the WJ-III, you were still able to score via an installable computer program that you purchased via one-time fee.
I know Pearson has really been making a push for the Q-Global and Q-Local scoring systems, which are both pay-per-score, although they still do offer score-it-yourself options for most (but not all) tests that I've seen. With the way the VA works, neither Q-Global or Q-Local would work (or likely ever be approved), and thus publishers stand to lose a large proportion of their customers if they ever convert 100% to cloud-based per-use scoring.
I'd imagine that like the WAIS and WMS, the WIAT is being phased into the Q scoring system, but may still have a one-time computer program available for purchase.
And I agree--Pearson's website is unnecessarily complex to navigate when it comes to finding pricing for the one-time purchase options.
Edit: Just checked, and it looks like Pearson still offers the manual scoring kit for the WIAT-III, but does not have computer scoring (unless you use Q-Global/Q-Interactive). Thus, if you don't want to pay per score, you'll need to score by hand.
Edit Again: I take that back, they offer unlimited-use annual scoring subscriptions, but also charge a per-use fee for reports (which I'm guessing are in more depth). And the manual scoring also includes reports. So maybe there's computer scoring after all.
Your best bet may be to call them directly; I'm already confused, and I haven't even looked at the WJ-III.