= Solution
Choose a coordinate ball $W$ around $q_1$ and a smaller convex coordinate ball $U$ whose closure lies in $W$. For $q_2\in U$, let $a$ be the constant coordinate vector from $q_1$ to $q_2$. Choose a <smooth bump function> $\rho$ supported in $W$ and equal to one on $U$, and define in the chart
$$
w=\rho a,
$$
extending it by zero outside $W$. This is a compactly supported smooth vector field. The segment $q_1+ta$ stays in $U$, where $w=a$, so uniqueness for ordinary differential equations gives
$$
\Psi^t(q_1)=q_1+ta,\qquad 0\leq t\leq1.
$$
In particular, $\Psi^1(q_1)=q_2$.
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