Take the vector field and extend it via $d\exp_{c(t)}|_\nu$, where $\nu$ is the normal bundle over $c$.
The tangent bundle of $c$, $Tc$, resides in $TM|_c$. The existence of a metric lets us pick out elements of $TM|_c$ perpendicular to $Tc$. All such perpendicular elements form the normal bundle $\nu(c\subset M)$. The exponential map takes a small neighborhood of the zero section of $\nu$ to a small open neighborhood of $c$.
If we have a vector field $X$ defined on $c$, we can extend it to a vector field on $\nu$, defined by extending it constantly on each fiber (note that $\dim\nu = \dim TM|_c$). Now just push $X$ out to $\exp\nu$ by $d\exp$, which is a diffeomorphism.
Alternately, since $c$ is a submanifold, it has an atlas of coordinate charts which take the form $c\times \mathbb{R}^{n-1}$ (here $n = \dim M$). You can extend $X$ to a neighborhood of $c$ by extending $X$ to a neighborhood of $c$ in each coordinate chart so that it commutes with the projection. (You will need to be a little careful about coordinate changes, though.)