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In appendix C of his textbook on general relativity, Sean Carroll introduces the notion of a set of vector fields defining an integral submanifold. He takes an $n$-dimensional manifold $M$, an $m$-dimensional submanifold $S$, and a set of $p$ linearly independent vector fields $V_{(a)}^\mu$, with $p \ge m$. Then he says that these vector fields "fit together to define $S$" if each vector is tangent to $S$ everywhere, so that the $V_{(a)}^\mu$'s span each tangent space $T_qS$. This then leads to a discussion of Frobenius' theorem.

But in the case that $p>m$, how can the vectors $V_{(a)}^\mu$ both be linearly independent and span $T_qS$, when the latter is $m$-dimensional? Does the author mean that it is enough that $T_qS$ is a subspace of the span of the vectors (at each given point $q \in S$)? Or are the vectors allowed to be linearly dependent at a given point, as long as they are not globally dependent? Or does he mean something else entirely?

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You're right that's a confusing paragraph; we cannot have more linearly independent vectors belonging to a subspace than its dimension; what he means is that merely that at each point, $T_xS$ is contained in the span of the vectors at that point (which is linearly independent).

Perhaps the following way of phrasing might make the setup of the problem clearer. We first introduce the notion of $p$-dimensional direction fields.

Definition $1$.

Let $1\leq p\leq n$ be integers, $M$ a smooth $n$-dimensional manifold. A $p$-direction field on $M$ is by definition a mapping $x\mapsto L_x$, which for each $x\in M$ assigns a certain $p$-dimensional vector subspace $L_x\subset T_xM$. We shall say the $p$-direction field $x\mapsto L_x$ is smooth if for each $x_0\in M$, there is an open neighborhood $U$ of $x_0$ in $M$ and smooth vector fields $\xi_1,\dots, \xi_p$ on $U$ (i.e $\xi_i:U\to TM$ is smooth and $\pi_{TM}\circ \xi_i=\text{id}_U$) such that for each $x\in U$, $\{\xi_1(x),\dots, \xi_p(x)\}$ is a basis for $L_x$.

You can obviously make the same definition for various levels of regularity: $C^0,C^1,C^2$ etc. So, really, the emphasis is not on the vector fields, rather it is on the subspaces $L_x$. The vector fields only come into play to define a notion of "smoothly varying family of subspaces".

Now we come to the definition of integral submanifolds. I'll list the two variants I've seen.

Definition $2$.

Let $1\leq q\leq p \leq n$ be integers, $M$ a smooth $n$-dimensional manifold, $x\mapsto L_x$ a smooth $p$-direction field on $M$. A $q$-dimensional (immersed) integral manifold of the direction field $x\mapsto L_x$ is by definition a pair $(N,f)$, where $N$ is a smooth $q$-dimensional manifold, and $f:N\to M$ is an immersion (i.e at each point $z\in N$, $Tf_z:T_zN\to T_{f(z)}M$ is injective) and is such that for each $z\in N$, the tangent mapping $Tf_z$ has image contained in $L_{f(z)}$ (in symbols $Tf_z(T_zN)\subset L_{f(z)}$).

The second variant doesn't deal with such immersions, but rather embedded submanifolds directly (which I find more intuitive).

Definition $2$'.

Let $1\leq q\leq p \leq n$ be integers, $M$ a smooth $n$-dimensional manifold, $x\mapsto L_x$ a smooth $p$-direction field on $M$. A $q$-dimensional (embedded) integral manifold is by definition a $q$-dimensional embedded submanifold $S\subset M$ such that for each $x\in S$, we have $T_xS\subset L_x$.

Frobenius' theorem is then about complete integrability, meaning it provides necessary and sufficient conditions such that given $L$, we can find $S$, where the dimensions are such that $q=p$ (not just $q\leq p$).

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