The title's question is not entirely accurate, as I had to cut down the number of characters to be under the maximum number of characters allowed. Here is my real question:
Let $M$ be a connected manifold with dimension $\geq 2$ and $p_1,\dots,p_k$ and $q_1,\dots,q_k$ be two collections of distinct points. Show that if $v_i \in T_{p_i}M - \{ 0 \}$ and $w_i \in T_{q_i}M - \{ 0 \}$, then there is a compactly supported isotopy $H$ from $\text{id}_M$ such that $H_1(x) = H (1,x)$ satisfies $DH_1|_{p_i}(v_i) = w_i$.
Because $M$ is connected with dimension $\geq 2$, I know that there exists a compactly supported isotopy $H: [0,1] \times M \to M$ that carries $p_i$ to $q_i$ for all $i$, or more precisely: $H(0,x) = x$ for all $x \in M$ and $H(1, p_i) = q_i$. However, I am not sure how to make it so that $DH_1|_{p_i}(v_i) = w_i$. Could you help? I have also read the answers from this post which confused me.
You say you already know how to find an isotopy $H$ that carries $p_i$ to $q_i$ for all $i$. The effect of this isotopy on vectors is that $H$ carries $v_i$ to some vector $u_i \in T_{q_i}M-\{0\}$, which is not necessarily the vector $w_i$ that you want.
So what remains is to follow up $H$ by some other isotopy which I'll call $G$, such that $G$ that leaves each $q_i$ stationary, and such that $G_1$ carries any given vector $u_i \in T_{q_i} M - \{0\}$ to any other given vector $w_i \in T_{q_i} M - \{0\}$ for $i=1,...,k$.
The linked answer shows how to do this if $k=1$. But what it actually shows is something a bit stronger: for each coordinate neighborhood $U_i$ of $q_i$ there exists an isotopy $G^i$ that is supported in $U_i$ and that carries $u_i$ to $w_i$.
To extend this to larger values of $k$ takes two tricks.
The first trick is to choose the coordinate neighborhoods $U_i$ carefully: since $q_1,...,q_k$ is a finite list of pairwise distinct points, we can use the Hausdorff property to find a pairwise disjoint collection of coordinate neighborhoods $U_1,...,U_k$, containing $q_1,...,q_k$ respectively. And now choose the respective isotopies $G^1,...,G^k$ as above.
The second trick is to chain the isotopies together: first do $G^1$, then $G^2$, and so on, finishing with $G^k$. Since these isotopies have pairwise disjoint supports $U_1,U_2,...,U_k$, you can write a single formula for this chained together isotopy: $$G(x,t) = \begin{cases} x &\quad\text{if $x \not\in U_1 \cup \cdots \cup U_k$} \\ x &\quad\text{if $x \in U_i$ and $0 \le t \le (i-1)/n$} \\ G^i(x,nt - (k-1)) &\quad\text{if $x \in U_i$ and $(i-1)/n \le t \le i/n$} \\ G^i(x,1) & \quad\text{if $x \in U_i$ and $i/n \le t \le 1$} \end{cases} $$ And now convince yourself that for each $i=1,...,k$ the following holds: as $t$ increases from $0$ to $(i-1)/n$ the vector $u_i$ is stationary; as $t$ increases from $(i-1)/n$ to $i/n$ the vector $u_i$ is carried to $w_i$; and as $t$ increases from $i/n$ to $1$ the vector $w_i$ is stationary. The net effect is that $G(\cdot,1)$ carries $u_i$ to $w_i$.