In DoCarmo’s Riemannian Geometry, a tensor of order $r$ on a Riemannian manifold is defined as a multilinear mapping $$T:\Xi^r(M)\rightarrow C^{\infty}(M)$$ where $M$ is a smooth manifold of dimension $n$, $\Xi(M)$ denotes the module of smooth vector fields on $C^{\infty}(M)$.
DoCarmo wants to show that $T$ is a pointwise object in the following sense: “Fix a point $p\in M$ and let $U$ be a neighborhood of $p$ in $M$ on which it is possible to define vector fields $E_1,...,E_n\in \Xi(M)$, in such a fashion that at each $q\in U$, the vectors $\{E_i(q)\}$,$i=1,...,n$, form a basis of $T_qM$. ...”
Does the author mean that for each neighborhood of $p$ there exists such vector fields, or what? If so, how can one define such vector fields?
Such fields do not exist on general neighborhoods of a point $p$ (since for example the whole manifold is a neighborhood of $p$). But for sufficiently small neighborhoods they do exist as the example of the coordinate vector fields of a chart containing $p$ shows.