I am trying to compute the vanishing of $\operatorname{Ext}^1$ for two sheaves of $\mathcal{O}_X$-Modules and I was wondering if it was possible to use some local argument to reduce the problem to the category of $\mathcal{O}_{X,x}$-modules.
So my question is: Are there any conditions on $\operatorname{Ext}^1(\mathcal{F}_x,\mathcal{G}_x)$ that are enough to show that $\operatorname{Ext}^1(\mathcal{F},\mathcal{G})=0$?
I am working in the case $X= \mathbb{P}^2$.
No. This is basically the point of sheaf cohomology -- global Ext is a global computation that relies on how the different pieces of the sheaf patch together.
For example, suppose $F$ and $G$ are coherent and locally free. Then $\mathrm{Ext}^i(F,G) = H^i(F^* \otimes G)$, whereas over any local ring $\mathcal{O}_{X,x}$, $Ext^i(F_x,G_x)$ vanishes for $i>0$ since $F_x$ is free.
That said, a statement of this form does hold for the sheaf version of Ext, namely
$\mathcal{E}xt(F,G)_x = Ext_{\mathcal{O}_{X,x}}(F_x,G_x)$
holds under the right finiteness hypotheses (coherent sheaves on a Noetherian scheme is enough, I think). So a necessary and sufficient for sheaf Ext to vanish is for each of these stalks to vanish.