Let, $\phi: X \rightarrow X$ be a diffeomorphism of a smooth compact manifold $X$ with no boundary. Let, $X_{\phi}$ be the quotient manifold $(X \times[0,1])/ \sim$ and $(x,1) \sim (\phi(x),0)$.
How do I come up with an example of a vector field(with no zeros) on such a manifold($X_{\phi}$)? And what about the Euler characteristic of $X_{\phi}$? I think it is $0$, but I'm not certain if my answer is correct.
You can just do something like $(x,t) \mapsto (0, \frac{\partial}{\partial t})$. Since it doesn't depend on $x$, it's compatible with your identification. Your manifold is almost $X \times S^1$, so this is almost like picking a nonvanishing vector field on $S^1$ and adding the zero section of $X$, in a way.
Since the manifold has a nonvanishing vector field, its Euler characteristic is zero.