Of course, we have that $p$ and $q$ can be connected by a path, since $M$ is locally Euclidean and therefore locally path-connected. My suspicion is that this statement, and the proof goes something as follows:
Let $\gamma:[0,1]\to M$ be a path from $p$ to $q$, and cover $\gamma([0,1])$ by open sets $U_1,\ldots,U_k$ where $f_i:U_i\to B(0,1)=\{x\in\mathbb{R}^n\big|\ |x|<1\}$ is a homeomorphism for all $i$, and such that there exist points $p_i\in U_i\cap U_{i+1}$ for all $1\leq i\leq k-1$, with $p_0=p,\ p_k=q$. Let $F_0:M\to M$ be the identity map, then we need only modify $F_0|_{U_1}:U_1\to U_1$ and extend to $M$ to get some homeomorphism $F_1:M\to M$ where $F_1(p_0)=p_1$. Once this is done, we proceed by basic induction, and let $F$ be the composition of the $F_i$.
However, I'm stuck on how to defined $F_1$ from $F_0$. I have some intuition about how to deform $U_1$, but I can't see how to formalize it, or how to ensure that ensure that $F_1$ is continuous.
I'm also curious if an idea like this holds for smooth manifolds as well.