One can define the algebra $A(K)$ of octonions over an arbitrary field $K$, see for example the command OctaveAlgebra in GAP: https://www.gap-system.org/Manuals/doc/ref/chap62.html . When $K$ is a finite field, this is a finite dimensional $K$-algebra and thus has finitely many elements. Let $A_q$ denote the octonions over a field with $q$ elemetns.

Question 1: What is the number of units in $A_q$? Can one even describe the (possibly non-associative) group of units up to isomorphism?

For $q=2$ the order is 120 and for $q=3$ the order is 4320. In both cases it is indeed a group according to GAP.

Question 2 is motivated by Will Sawin's comment (I forgot the unit "group" might not be associative):

Question 2: For which $q$ is the unit "group" of $A_q$ associative?

It would be interesting to see what the smallest $q$ is such that the unit "group" is not associative.