This is Vakil 13.7 E, self-study.
We are to show that if $X$ is a scheme and $\mathcal F$ is a finite type quasicoherent sheaf on $X$, then if $p \in U \subset X$ is an open neighborhood of $p$ and $a_1, … , a_n \in \mathcal F(U)$ have images generating the fiber $\mathcal F_p \otimes \kappa(p)$, then there must be an affine open neighborhood $p \in \operatorname{Spec} A \subset U$ such that the $a_i$ each restricted to $\operatorname{Spec} A$ generate $\mathcal F(\operatorname{Spec}A)$ as an $A$-module, and for each $q \in \operatorname{Spec} A$, the (images of) the $a_i$ generate $\mathcal F_q$ as an $\mathcal O_{X, q}$-module.
Here is my attempt, but something feels off about it:
If we assume $U$ is already an affine open $\operatorname{Spec}A$, then we know $\mathcal F$ is locally a finite type $A$-module $M$ on $U$. Then the fiber at $p$ is isomorphic to $M_p/pM_p$. Since being a finite type $A$-module is a local property, $M_p$ is a finite type $A_p$-module. Since $p$ is a prime ideal in $A$, by version 8 of Nakayama's Lemma from the Stacks Project's tag 07RC, $M_p$ is generated by the images of the $a_i$. Since $p$ was arbitrary, again by the localness of being finite type, $M$ is finitely generated by the $a_i$.
Using localness once more, $M_q$ is generated by the images of the $a_i$ for any $q \in \operatorname{Spec}A$.
Something about assuming $U$ was affine feels off, almost like I did not quite show what was asked. Also, we did not show that finite type was local in the notes thus far, only that $M$ is finite type over $A$ if and only if $M_{f_i}$ is finite type over $A_{f_i}$, where the $f_i$ generate $A$. I am not sure this allows me to conclude the same about localizing at a prime. It also feels like I used localness "too much."
Best Answer
You have already reduced to the affine case (and there is nothing wrong with your reduction) so I will write things purely in terms of rings and modules.
We are in a situation where we have a ring $A$, a finitely-generated $A$-module $M$, and elements $a_1,\dots,a_n\in M$ whose images generate $M_p$ as an $A_p$-module. Our goal is to find some $g\in A\smallsetminus p$ for which the images of $a_i$ generate $M_g$ as an $A_g$-module, because then $\operatorname{Spec}(A_g)$ is the affine open neighborhood of $p$ you are looking for.
Now you should use the fact that $M$ is finitely generated over $A$ and write down a generating set, say $x_1,\dots,x_m$, and notice that the $x_i$ will generate $M_g$ over $A_g$ for any $g$ we choose. Now we know that the $a_i$ generate $M_p$ over $A_p$, so for each $i$ we can write $$x_i=\sum_j \big(\frac{b_{ij}}{s_{ij}}\big)a_j\:\:\:\:\:\:\:\:\text{in $M_p$}$$ for some elements $b_{ij}/s_{ij}\in A_p$, so $s_{ij}\in A_p\smallsetminus p$. I claim we should take $g:=\prod_{i,j}s_{ij}$; since the $x_i$ already generate $M_g$ over $A_g$, you just need to verify that each of these can be generated with the $a_i$ in $M_g$, and this is easy to see by our choice of $g$.
Edit: the comments below are correct, one needs a tweak to the argument above that I'm too lazy to type out right now.