= Solution
A <sheaf of modules> $\mathcal F$ over $\mathcal O_X$ is <quasi-coherent> when locally it admits a presentation
$$
\mathcal O_X^{(J)}\longrightarrow\mathcal O_X^{(I)}\longrightarrow\mathcal F\longrightarrow0,
$$
with arbitrary indexing sets $I,J$. On an affine chart, sheafifying the corresponding <module> presentation and using <exactness of localization> identifies its cokernel with a <sheaf> $\widetilde N$. Thus one may equivalently require that locally on affine charts $\mathcal F$ is associated with a <module>. This does not impose finite generation, which belongs to the stronger coherent condition.
On the affine variety $X$ with <coordinate ring> $A$, put $M=\Gamma(X,\mathcal F)$. Refine local <module> charts to a finite principal <open cover> $X=\bigcup_jD(g_j)$ such that $\mathcal F|_{D(g_j)}\cong\widetilde N_j$ for an $A_{g_j}$-module $N_j$. This is possible because principal opens form an <open basis> and $X$ is <quasi-compact>. On the overlap $D(g_jg_k)$ the sections are the corresponding further <localization> of $N_j$ or $N_k$. The <sheaf gluing axiom> gives an equalizer
$$
0\longrightarrow M\longrightarrow\prod_jN_j\longrightarrow\prod_{j,k}\mathcal F(D(g_jg_k)),
$$
where the last map is the difference of restrictions. For any $f\in A$, localize this sequence. <Exactness of localization> and commutation with finite products give exactly the equalizer for the cover $D(f)=\bigcup_jD(fg_j)$. Hence
$$
\boxed{M_f\cong\Gamma(D(f),\mathcal F).}
$$
The <isomorphisms> are canonical and commute with all basic-open restrictions. Since the sections of $\widetilde M$ on these opens are $M_f$, they identify the two <sheaves>:
$$
\boxed{\mathcal F\cong\widetilde{\Gamma(X,\mathcal F)}.}
$$
Conversely $\widetilde M$ is quasi-coherent for every $A$-module $M$, since a free presentation of $M$ gives the required <sheaf> presentation. Also $\Gamma(X,\widetilde M)=M$, and <module> homomorphisms sheafify, while a <sheaf> morphism is determined on every $D(f)$ by the <localization> of its map on <global sections>. This proves the <affine module-sheaf equivalence>.
For a short exact sequence of <quasi-coherent sheaves>, the sequence of their <stalks> is exact. Under this equivalence the <stalk> at $x$ is the <localization> of the global-section <module> at $\mathfrak m_x$. Exactness of <module> sequences can be checked at all maximal ideals, by <localization detects zero elements> applied to their homology, so the global-section <modules> also form a short exact sequence. Therefore \b[<global sections> are exact on <quasi-coherent sheaves> over an affine variety]. This is stronger than the left exactness of the <global section functor> on arbitrary <sheaves>, and follows from <localization>, not from assuming the cohomology vanishing still to be proved.
For an open inclusion $j:U\hookrightarrow X$, the requested <sheaf> is
$$
\boxed{{}_U\mathcal F=j_*(\mathcal F|_U),\qquad({}_U\mathcal F)(V)=\mathcal F(V\cap U).}
$$
Restriction gives $\mathcal F\to{}_U\mathcal F$. This <direct image from an open restriction> is not extension by zero: its <stalks> outside $U$ may be nonzero.
The <locally vanishing principle for sheaf cohomology> says that a class $\xi\in H^i(X,\mathcal F)$, $i>0$, is killed on a suitable neighbourhood of every point. Under the stated hypothesis that basis opens and their finite intersections have zero cohomology in degrees $1,\ldots,i-1$, these neighbourhoods can be chosen in that basis so that the image of $\xi$ under
$$
H^i(X,\mathcal F)\longrightarrow H^i(X,{}_U\mathcal F)
$$
is zero. In particular its restriction in $H^i(U,\mathcal F|_U)$ is zero. This is a statement about individual classes; it does not assert that every locally vanishing class is already globally zero.
Here is a noncircular proof of <vanishing of quasi-coherent cohomology on an affine scheme>, applied to the variety. Induct on $i>0$, simultaneously for every affine variety and <quasi-coherent sheaf>. Assume all lower positive degrees vanish. The principal-open basis is closed under finite intersections, so it satisfies the principle's hypothesis. Given $\xi\in H^i(X,\mathcal F)$, choose a finite principal cover $\mathcal U=(D(f_j))$ on which it restricts to zero.
Applying the <Čech cochain complex> to a <flasque resolution> gives the <Čech lifting below the first possible local cohomology degree> comparison segment, under lower-degree vanishing on all intersections
$$
0\longrightarrow\check H^i(\mathcal U,\mathcal F)\longrightarrow H^i(X,\mathcal F)\longrightarrow\prod_jH^i(D(f_j),\mathcal F).
$$
For completeness, the double complex has terms $\check C^p(\mathcal U,\mathcal I^q)$. Its augmented rows are exact because each $\mathcal I^q$ is flasque, so its total cohomology is the cohomology of <global sections> of the resolution. The vertical cohomology on intersections in degrees $0<q<i$ is zero. Equivalently, start with local primitives of a cocycle representing $\xi$, take their differences on pairwise overlaps, and solve successively for primitives of those differences in degrees $i-1,i-2,\ldots,1$. The last difference is a Čech $i$-cocycle with values in $\mathcal F$. This identifies the kernel of the restriction map with the displayed Čech group. No vanishing in degree $i$ on the intersections has been assumed.
It remains a <module> calculation. Write $\mathcal F=\widetilde M$. The augmented Čech complex is
$$
0\longrightarrow M\longrightarrow\prod_jM_{f_j}\longrightarrow\prod_{j<k}M_{f_jf_k}\longrightarrow\cdots.
$$
Since the $D(f_j)$ cover $X$, the $f_j$ generate the <unit ideal>. This complex is exact. To see it algebraically, use alternating cochains, with repeated indices giving zero. For a cocycle $c$, choose $N$ large enough to clear every $f_j$-denominator in $f_j^Nc_{jI}$, as well as the finitely many cocycle relations; <vanishing criterion in a module localization> allows a further power to clear relations which initially hold only after <localization>. Since the ideal $(f_j^N)$ is still the <unit ideal>, choose $a_j$ with $\sum_ja_jf_j^N=1$. Define
$$
b_I=\sum_j a_jf_j^Nc_{jI},
$$
using those cleared representatives in $M_{f_I}$. The cocycle identity gives $\delta b=(\sum_ja_jf_j^N)c=c$. The same argument with the augmentation gives gluing and uniqueness in degree zero. This is <exactness of the unit-ideal localization Čech complex>.
Thus $\check H^i(\mathcal U,\mathcal F)=0$, so the class $\xi$, whose restrictions were zero, is zero. The induction starts at $i=1$, when the lower-degree condition is empty. We have proved
$$
\boxed{H^i(X,\mathcal F)=0\quad\text{for every }i>0\text{ and every affine quasi-coherent }\mathcal F.}
$$
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