Not by default, no.
Things are unlocked like this:
You get Radicals to Guru and you unlock Kanji lessons.
You get Kanji to Guru and you unlock Vocab lessons.
So when you hit a new level, the following things are unlocked:
The new radical and kanji for that level.
The vocabulary lesson for the kanji you got to guru and caused you to level up.
Because only the radical and kanji are required to Guru level before you level up, it’s entirely possible (if you change your settings) to level up without doing vocabulary lessons at all. Though this is a Bad IdeaTM of course, since the vocabulary lessons exist to reinforce the kanji you’ve learned. So the point I was making about lesson order is this; if a person wants to level up at the maximum speed (6d 20hr) then doing the radical lessons first makes the most sense, because they control the unlock of the rest of the kanji for a level.
I’m not in any way saying that a person SHOULD do that, because it’s entirely down to personally preference and goals. The biggest truth of success on wanikani is doing stuff at a pace that suits you and if that means doing 10 lessons a day, that’s fine.
By default though, and as things currently stand if you’ve never touched any settings, things are kept on the default lesson ordering settings (By Level, then By Subject) and you’ll never see stuff from a new level before you’ve finished the stuff from the current level. If things were set to Shuffled, for example, and you were doing 10 lessons a day, you’d theoretically start getting stuff from the new level before you finished items from the previous.