Given a manifold $M$ and a distribution $D \subset TM$, an integral manifold of $D$ is defined as a nonempty immersed submanifold $N \subset M$ such that $T_pN = D_p$ for all $p \in N$.
Why are we not able to define an integral manifold as an embedded submanifold? My guess is that embeddings are too strict, but is there an intuitive geometric reason behind this?
If you just want to define integrability of distributions, then you can equivalently use embedded submanifolds. (Indeed, for an immersed submanifold $N\to M$ and $x\in N$ there always is an open neighborhood of $x$ in $N$ which is an embedded submanifold in $M$.) But if you go to the general study of integral submanifolds (usually called the global version of the Frobenius theorem), then embedded submanifolds are too restrictive. What you prove there is the through each point there is a maximal connected (immersed) integral submanifold, called the leaf through $x$. This is not embedded in general, but as stated in the comment of @Ivo_Terk, it can be shown that these leaves are always weakly embedded or so-called initial submanifolds. But this is the result of a theorem (and it is a global condition) and it would be rather strange (and not intuitive) to make this part of the definition.