Not an answer, but a pointer to the difficulties:

If you restrict yourself to just one dimensional distributions, you immediately see a problem.

Let $V$ be a continuous (non-vanishing) vector field on $\mathbb{R}^n$, $n > 1$, it clearly generates a $C^0$ sub-bundle of the tangent bundle. By Peano's existence theorem we have that $V$ has *at least one* integral curve passing through any initial point. On the other hand, contrasted against Picard's existence theorem which requires Lipschitz regularity of the vector field (or in your case, $C^1$ suffices), this integral curve is not guaranteed to be unique.

Why are curves important? Frobenius in $C^1$ can be interpreted in the "integral form" as the following: To be sure that the distribution integrates to a submanifold, if you travel infinitesimally first in the $V_1$ direction, then in the $V_2$ direction, and compare with traveling first in the $V_2$ direction, then in the $V_1$ direction, the difference should be in a direction that is inside the subbundle. (In other words, you cannot move off the manifold by moving inside the manifold.)

So to have any hope of having a Frobenius like statement you need to overcome this problem of having way too many integral curves. This, in fact, also causes problems for the existence of a foliation!

An example:

Consider the vector field $V$ on $\mathbb{R}^2$ given by
$$ V(x,y) = (1, \sqrt{|y|}) $$
Integral curves through $(0,0)$ include
$$ \gamma_{0}(t) = (t,0)\qquad \gamma_{+}(t) = (t,t^2 / 4) \qquad \gamma_{-}(t) = (t,-t^2/4) $$
for positive time $t$, and only $$\gamma(t) = (t,0)$$ is admissible for negative time $t$. (In fact, one can also transition from $\gamma_0$ to a suitably translated $\gamma_{\pm}$ at any $t \geq 0$.)
This shows that every integral curve of $V$ must become tangent to the $x$ axis in finite negative time. Hence it is impossible to obtain a foliation to this non-vanishing, continuous vector field.

Thus you see that the issue of regularity occurs when just one dimension is considered, much before the statement of Frobenius theorem, which is a compatibility condition on multiple dimensions, comes into play.