I am reading "Calculus on Manifolds" by Michael Spivak.
The author defined a vector field as follows:
To be precise, a vector field is a function $F$ such that $F(p)\in\mathbb{R}_p^n$ for each $p\in\mathbb{R}^n$.
What is the range of a vector field?
Is the range of a vector field $\bigcup_{p\in\mathbb{R}^n}\mathbb{R}_p^n$?
For whatever reason, we do not really use the word "range" in math after calculus or so in English-speaking countries, preferring "co-domain" or "target" to mean the thing the math hypothetically lands in and "image" to mean the stuff that gets hit by the map.
Assuming you mean co-domain, then yes, your interpretation of Spivak's setup is right, although it would be more precise to write $\bigsqcup_X \mathbb{R}_p^n$ to emphasize that this is a disjoint union. Note also that the union should be "over" $X$ or whatever the vector field is "on," not $\mathbb{R}^n$. Note that this is much much bigger than the image, since a point $p$ must map to an element of $\mathbb{R}_p^n$ (same $p$, not some other $q$).