Show that there exists a smooth map $F: M \to M$ that is homotopic to the identity and has no fixed points.

Your argument is good. Here's how I'd pick appropriate $V_p$ and $t_p$:

Fix coordinates $(x,y^1,y^2,\ldots)$ on $U_p$ such that $\partial/\partial x = V$. Since $U_p$ is a neighbourhood of $p$, it contains some coordinate ball $B_r(p)$. Since flow solutions are unique, $\theta(t,x,y) = (x+t,y)$ is true whenever $(x,y)$ and $(x+t,y)$ are both in the coordinate chart domain; and by the triangle inequality we know $(x+t,y) \in B_r(p)$ whenever $|t|<r/2$ and $(x,y)\in B_{r/2}(p).$ Thus we can take $V_p = B_{r/2}(p)$ and $t_p = r/2$.