Suppose $\mathcal{F}$ is a skyscraper sheaf supported on $\bar{\{\mathfrak{p}\}}$, the stalk is $M$, What is its global section over $\operatorname{Spec} A$?
We need to find a module $N$ such that $N_{\mathfrak{q}}=M$ when $\mathfrak{q}\in V(\mathfrak{p})$ and $N_{\mathfrak{q}}=0$ otherwise.
Is the module $N=M$? If so, why does it satisfy the conditions mentioned?
Suppose we take the module $M=A$ the conditions are not satisfied.
Let $X$ be a scheme and $x\in X$ be a point.
The problem you ask about is more subtle than it seems: if you want a skyscraper sheaf $\mathscr F$ on $X$ to be quasi-coherent, you have to assume that $\mathscr F$ is a sheaf of $\mathscr O_X$-Modules.
However skyscraper sheaves are usually assumed (in Hartshorne for example) to be only sheaves of abelian groups.
Here is how to define skyscraper sheaves which are $\mathscr O_X$-Modules:
Start with an $\mathscr O_{X,x}$-module $M$ and consider the one point locally ringed space $P=(\{x\},\mathscr O_{X,x})$.
The module $M$ can be considered as a quasi-coherent sheaf $P$.
Now, there is a natural morphism of locally ringed spaces $P\to X$ and you may define the $\mathscr O_X$-Module $\mathscr F=j_\ast M$: this is the required skyscraper sheaf at $x$ constructed from the $\mathscr O_{X,x}$-module $M$.
In the special case $X=\text {Spec} (A)$ you ask about the $A$-module of global sections of $\mathscr F$.
It is given by the formula $\Gamma (\text {Spec} (A),\mathscr F)=M$.
There is however again a subtlety: the product of the global section $m\in M=\Gamma (\text {Spec} (A),\mathscr F)$ by the global function $a\in A=\Gamma (\text {Spec} (A),\mathscr O)$ is $a_x\cdot m\in M$: you have to remember that $M$ is an $\mathscr O_{X,x}$-module and the multiplication of $m\in M$ by $a_x\in \mathscr O_{X,x}$ thus makes sense.
Edit: The skyscraper sheaf $\mathscr F$ is not necessarily quasi-coherent
As a counterexample take for $X$ the affine line $\mathbb A^1_k=\text {Spec}(k[T])$ over a field $k$, for $x$ the origin $O$ of $X$ (corresponding to the maximal ideal $(T)$) and for $M$ take the field $k(T)$ seen as a module over $\mathscr O_{X,x}=\mathscr O_{X,O}=k[T]_{(T)}$.
Then the associated skyscraper sheaf $\mathscr F$ is not quasi-coherent: if it were we would have for the sections of $\mathscr F$ over $X\setminus \{O\}=D(T)$ the equality $\Gamma (D(T),\mathscr F)=k(T)\otimes _{k[T]} k[T, T^{-1}]=k(T)$ , whereas in reality $\Gamma (D(T),\mathscr F)=0$
If you restrict to schemes over $k$ for some fixed field $k$, and consider skyscraper sheaves of $k$-modules (rather than just abelian groups), then I think that the following slight variation on Georges' answer gives a way of viewing skyscraper sheaves as quasi-coherent sheaves.
Let $(X,\mathcal O _X)$ be a locally ringed space over $k$ and $x \in X$. Assume that the residue field at $x$ is isomorphic to $k$, i.e. that for every $x \in X$ the canonical morphism
$$ k \to \mathcal O _{X,x} \to \mathcal O_{X,x} / \mathfrak m _x $$
is an isomorphism, we then have a well defined map
$$\mathrm{ev}_x : \mathcal O_{X,x} \to k$$
Instead of $(\{x\},\mathcal O _{X,x})$ take the one point ringed space $(\{x\},k) = \mathrm{Spec}(k)$. Note that we have
$$ \mathcal O _{\{x\}} \text{-mod} = \text{Sh}_{k \text{-mod}} (\{x\}) = k \text{-mod}$$
Then $\mathrm{ev}_x$ gives a morphism of locally ringed spaces
$$(i_x,\mathrm{ev}_x): (\{x\},k) \to (X,\mathcal O_X)$$
and functors
\begin{align*} f^{-1} : \text{Sh}_{k\text{-mod}} (X) & \to k \text{-mod} \\ f_\ast : k \text{-mod} & \to \mathcal O_X \text{-mod} \end{align*}
For any sheaf $M$ of $k$-modules on $X$ we then have the $\mathcal O_X$-module $f_\ast f^{-1} M$.
If $M$ is a skyscraper sheaf of $k$-modules supported at $x$, then $f_\ast f^{-1} M \cong M$ as sheaves of $k$-modules, but $f_\ast f^{-1} M$ is an $\mathcal O_X$-module with action
$$ a \cdot m = (\text{ev}_x (a_x)) m $$
for any open $U \subset X$ containing $x$ and sections $a \in \mathcal O_X(U)$ and $m \in M(U)$.
If $(X,\mathcal O_X)$ is a scheme then since $(\{x\},k)=\text{Spec}(k)$ is a scheme (unlike $(\{x\},\mathcal O_{X,x})$ ), we have that $f_\ast f^{-1} M$ is quasi-coherent because it is the pushforward of the quasi-coherent sheaf $f^{-1}M$.
For example if $k$ is algebraically closed and $(X,\mathcal O_X)$ is locally of finite type over $k$, then the construction above gives you a way to view any skyscraper sheaf of $k$-modules supported at a closed point as a quasi-coherent sheaf.
A late answer, but this criterion is exercise 6.8.3 in Bosch's Algebraic geometry and commutative algebra:
Let $X$ a scheme, closed point $x$, and $\mathcal O_{X,x}$-module $F$. There is a unique associated skyscraper sheaf structure $\mathcal F$ such that the stalk $\mathcal F_y=0$ for $y\neq x$ and $\mathcal F_x=F$. The sheaf $\mathcal F$ is canonically an $\mathcal O_X$-module, and it is quasicoherent iff $F_y=0$ for every $x\neq y\in \text{Spec }\mathcal O_{X,x}$.
I'll give an answer in the special case of the skyscraper sheaf $\mathbb C_p$ on $\mathbb P^1_\mathbb{C}$. The answer is yes. For convenience, take $p=0$. We know $\mathbb C_p$ fits into the below exact sequence: $$ 0 \to \mathcal O_X(-p) \to \mathcal O_X \to \mathbb C_p \to 0 $$ If we restrict to $\mathbb A^1$, this sequence looks like: $$ 0 \to (t \cdot k[t])^\sim \to k[t]^\sim \to (\frac{k[t]}{t k[t]})^\sim \to 0 $$ One can verify the stalks of the third term are $0$ away from the origin, and at the origin, isomorphic to $\mathbb C$ with $\mathbb C[t]_{(t)}$-module structure given by $t$ acting as multiplication by $0$. On the other $\mathbb A^1$ patch of $\mathbb P^1$, $\mathbb C_p$ is given by the sheafification of the zero module. Hence $\mathbb C_p$ is a quasi-coherent sheaf.
