Let's agree that whatever "supersymmetry" means it has something to do with working in the symmetric monoidal category of super vector spaces (e.g. we might want to consider Lie algebras or commutative algebras in this category), or something like it. The question is what, if anything, this has to do with the sphere spectrum.

Here is at least the beginning of the story as I understand it. The sphere spectrum has the following universal property: it is the free spectrum, or the free infinite loop space, on a point. Said in a more explicitly higher categorical way,

The sphere spectrum $\mathbb{S}$ is the free symmetric monoidal $\infty$-groupoid with inverses on a point.

Okay, but $\infty$ is a pretty big number, so let's shield ourselves from the full power of this result by $n$-truncating it. Now it says

The $n$-truncation $\Pi_{\le n}(\mathbb{S})$ of the sphere spectrum is the free symmetric monoidal $n$-groupoid with inverses on a point.

Now let's specialize this result to small values of $n$.

$n = 0$ is easy and familiar: the $0$-truncation of the sphere spectrum is (the zeroth stable homotopy group of spheres, which is) $\mathbb{Z}$, and its universal property is that it is the free abelian group on a point.

$n = 1$: the $1$-truncation of the sphere spectrum is a symmetric monoidal groupoid with $\pi_0 \cong \mathbb{Z}$ and $\pi_1 \cong \mathbb{Z}_2$, and its universal property is that it is the free symmetric monoidal groupoid with inverses (sometimes called a "Picard groupoid") on a point. This means that if $C$ is any other symmetric monoidal category (we'll be working with its maximal subgroupoid) and $c \in C$ is an invertible object in $C$, then there's a canonically defined symmetric monoidal functor $\mathbb{S} \to C$ which takes the value $c$ on the generator $1 \in \pi_0(\mathbb{S})$. On objects it sends $n \in \pi_0(\mathbb{S})$ to $c^{\otimes n}$, while on morphisms it sends $-1 \in \pi_1(\mathbb{S})$ to the "sign" of $c$, namely the value of the braiding

$$\beta_{c, c} : c \otimes c \to c \otimes c$$

regarded as an element of $\text{Aut}(c \otimes c) \cong \text{Aut}(1)$, where $1$ is the tensor unit (this identification is canonical given that $c$ is invertible). Notably, this is equal to $-1$ on the odd invertible super vector space and $1$ on the even invertible super vector space. (Several other descriptions of how $-1 \in \pi_1(\mathbb{S})$ acts are possible: for example, it can also be described as $\text{tr}(\text{id}_c)$. This comes from the cobordism hypothesis.)

From here it's possible to give a universal property of a version of super vector spaces: namely,

The symmetric monoidal category of $\mathbb{Z}$-graded vector spaces over a field $k$ equipped with the Koszul sign rule is the free symmetric monoidal cocomplete $k$-linear category on an invertible object of sign $-1$.

One might then hope to, for example, define higher analogues of super vector spaces by increasing the category number from here. Ganter and Kapranov also use the $2$-truncation $\Pi_{\le 2}(\mathbb{S})$ to define higher analogues of the sign character and hence of symmetric and exterior powers here.

However, there are reasons to believe this is not really what's going on. Super vector spaces are arguably not interesting because of how maps out of them behave but because of how maps into them behave: namely, Deligne's theorem about nice symmetric monoidal categories admitting a fiber functor to super vector spaces. This theorem can be intepreted as saying that the symmetric monoidal category of super vector spaces over $\mathbb{C}$ is in some sense "algebraically closed" (compare: every finite-dimensional commutative $k$-algebra admits an algebra homomorphism to the algebraic closure of $k$), or even the "algebraic closure" of, say, the symmetric monoidal category of vector spaces over $\mathbb{R}$.

So a different claim about what "supersymmetry" really is is that it's the study of categorified algebraic closures in this sense. Notably, the analogue of the theorem above with $\mathbb{R}$ replaced by other fields fails in positive characteristic: a conjecture about the correct replacement is given by Ostrik here. I learned this from Theo Johnson-Freyd; see, for example, this paper, where he gives three conjectures for how this theory of categorified algebraic closures of $\mathbb{R}$ generalizes to higher category number, only one of which is related to the sphere spectrum.