It is not really a question of inner forms. What happens is that the algebraic group $G_2$ has an extra endomorphism $\varphi$ whose square is the Frobenius map (over the appropriate finite field). Just as for any algebraic group over a finite field $F$ its rational points over $F$ are the fixed points of the Frobenius endomorphism the Suziki groups are, by definition, the fixed points of $\varphi$. Again, just as the Frobenius, on points over the algebraic closure of $F$ $\varphi$ is an automorphism of the abstract group. However, that is misleading, the essential points is that it is an endomorphism (which definitely is not an automorphism) of the algebraic group. Most of the properties of points over $F$ of a semi-simple algebraic group $G$ defined over $F$ follows from the algebro-geometric theory of $G$ and the properties of the Frobenius endomorphism. Similarly, most of the properties of Suziki groups follows from the algebro-geometric theory of $G_2$ together with the properties of $\varphi$. As $\varphi$ is very similar to the Frobenius endomorphism this works almost the same way as if $\varphi$ were indeed a Frobenius endomorphism.
Addendum: As one simple example of the similarity of $\varphi$ to a Frobenius consider the problem of computing the order of the Suzuki groups. As the square of $\varphi$ is the Frobenius, the action of it on the tangent space at any fixed point is nilpotent. This implies that such a fixed point appear with multiplicity one in the Lefschetz fixed point formula and the order of its group of fixed points is thus equal to the Lefschetz trace on (étale) cohomology of the algebraic group $G_2$. That cohomology can be canonically expressed in terms the action of the Weyl group on the character group of the maximal torus (see for instance example in SGA 4 1/2) and how $\varphi$ acts on that character group is essentially part of the definition of $\varphi$.