I am not sure how it works in Japanese, but in Chinese there were certain rules about the stroke order that made it so you didn’t have to memorise the stroke order for every word. Once you get used to it (doesn’t take long, few weeks to internalise maybe), you’ll do the majority of new characters the right way without seeing stroke order, and the minority that deviates is done wrong by many native speakers as well (bad excuse to do it wrong yourself, but ah well…). Thus it is important to learn stroke order rules! Not the stroke order of individual characters so much.
A quick google shows a guide on the rules for Japanese here on Tofugu: Kanji Stroke Order: How to Guess it Every Time, that seems to be similar to the Chinese rules.
Stroke orders also very much follow radicals (though not always exactly…). But be aware here that what Wanikani calls radicals is not exactly the same as the established radical sets! It’s more akin to what is traditionally called components.
The majority of Chinese characters are so-called picto-phonetic characters, that have 1 or 2, sometimes even 3 radicals that say something about the meaning, and then another radical to give the pronunciation (a reading radical, as you might say). As I just started Wanikani and learning hanzi, I can’t say if it’s the same with on’yomi readings of hanzi or if it has changed too much over time. Actually, Ilooked it up, 61 of jouyou kanji is of the ‘semantic-phonetic type’:
Radicals are written with normally 1-3 components. In general use you find roughly 500 components with Chinese characters, not sure about the kanzi. There’s about 200 radicals that make up nearly all characters, of which you already get very far if you know the 100 most frequent radicals.
Radicals are often indexed, as they are used for dictionaries, making fonts, etc.
In Japan a more expansive set of 214 and a base set of 74 radicals is normally used. If you know the stroke order of the base 74 radicals (of which only 49 are used in the jouyou kanji, in case you stick to them for now), you will get VERY far in writing the stroke order of all kanzi properly. I’ll bet a keg on that. Knowing those 74 radicals also allows full use of some dictionaries (others use the more expansive set of 214 radicals).
Components don’t really have such official indexes, and what wanikani calls radicals is in reality a proprietary set of components and not radicals. Wanikani is teaching the reading of kanji, and has specifically designed their unique index of components for it (which they call radicals). That is great for learning to read the kanji, but is not an ideal basis for learning other things like writing and stroke order.
So if you want to move beyond wanikani, and learn stroke order as well as other things:
- Learn the stroke order rules as per the tofugu article above.
- Practice using the common radicals with their stroke order! Wanikani doesn’t differentiate much from them (many you will already know as a kanji actually, many radicals can on their own also be a kanji, and many ‘wanikani radicals’ are also equal to a kanji), so I bet that if you are far into Wanikani, that you will know the meaning and on’yomi reading of nearly all useful radicals (wikipedia is your friend in finding them, see how many you already know for everything but stroke order: https://en.wikipedia.org/wiki/List_of_kanji_radicals_by_frequency).
Then, going the proper route seems worth the effort (rote learning of individual kanji stroke order). Otherwise, I’d say no: just learn the general principles of order, and that should be plenty.