The integral formula for the Moyal/Groenewold $\star$ product in Ref. 1 reads
$$ (f \star g)(y)~=~\int_{\mathbb{R}^4} \! \frac{d^2u~d^2v}{(2\pi\hbar)^2}~
\exp \left[\frac{i}{\hbar} u_{\alpha} \epsilon^{\alpha\beta} v_{\beta}\right]
f(y+u)g(y+v).\tag{1}$$
Remark: The argument $u_{\alpha} \epsilon^{\alpha\beta} v_{\beta}$ inside the exponential of the integral formula (1) has an interesting geometric interpretation as (twice) a signed area of a triangle in phase space $\mathbb{R}^2$, cf. Ref. 2.
Usually the Moyal/Groenewold $*$ product is defined as
$$ (f \star g)(y)~:=~ f(y) \exp \left[\stackrel{\leftarrow}{\frac{\partial}{\partial y_{\alpha}}} i\hbar\epsilon_{\alpha\beta} \stackrel{\rightarrow}{\frac{\partial}{\partial y_{\beta}}} \right] g(y). \tag{2}$$
Let us for completeness prove$^1$ the integral formula (1) from the definition (2) for a pair of sufficiently well-behaved functions $f$ and $g$:
$$\begin{align}(f \star g)(y)~\stackrel{(2)}{:=}~~~& f(y) \exp \left[\stackrel{\leftarrow}{\frac{\partial}{\partial y_{\alpha}}} i\hbar\epsilon_{\alpha\beta} \stackrel{\rightarrow}{\frac{\partial}{\partial y_{\beta}}} \right] g(y) \cr
~=~~~&\int_{\mathbb{R}^4} \! d^2u~d^2v ~\delta^2(u)~\delta^2(v) \exp \left[\frac{\partial}{\partial u_{\alpha}} i\hbar\epsilon_{\alpha\beta} \frac{\partial}{\partial v_{\beta}}\right]
f(y+u)g(y+v)\cr
~=~~~&\int_{\mathbb{R}^8} \! d^2u~d^2v~\frac{d^2k}{(2\pi)^2}\frac{d^2q}{(2\pi)^2}
\exp i\left[k\cdot u+q\cdot v+\frac{\partial}{\partial u_{\alpha}} \hbar\epsilon_{\alpha\beta} \frac{\partial}{\partial v_{\beta}}\right]
f(y+u)g(y+v)\cr
~\stackrel{\text{IBP}}{=}~~&\int_{\mathbb{R}^8} \! d^2u~d^2v~\frac{d^2k}{(2\pi)^2} \frac{d^2q}{(2\pi)^2}
\exp i\left[k\cdot u+q\cdot v-k^{\alpha} \hbar\epsilon_{\alpha\beta} q^{\beta}\right]
f(y+u)g(y+v)\cr
~=~~~&\int_{\mathbb{R}^6} \! d^2u~d^2v~\frac{d^2q}{(2\pi)^2} ~\delta^2(u_{\alpha}-\hbar\epsilon_{\alpha\beta} q^{\beta}) ~e^{i q\cdot v}
f(y+u)g(y+v)\cr
~\stackrel{(6)}{=}~~~&\int_{\mathbb{R}^6} \! d^2u~d^2v~\frac{d^2q}{(2\pi\hbar)^2} ~\delta^2( q^{\beta}-u_{\alpha}\epsilon^{\alpha\beta}/\hbar ) ~e^{i q\cdot v}
f(y+u)g(y+v)\cr
~=~~~&\int_{\mathbb{R}^4} \! \frac{d^2u~d^2v}{(2\pi\hbar)^2}~
\exp \left[\frac{i}{\hbar} u_{\alpha} \epsilon^{\alpha\beta} v_{\beta}\right]
f(y+u)g(y+v).\end{align}\tag{3}$$
The functions $f$ and $g$ should be sufficiently well-behaved in order for the above integrals and manipulations (3) to make mathematical sense. Let us
here give a necessary and sufficient condition for the integrand
$$ h(u,v) ~:=~\exp \left[\frac{i}{\hbar} u_{\alpha} \epsilon^{\alpha\beta} v_{\beta}\right]
f(y+u)g(y+v), \qquad u,v,y ~\in ~\mathbb{R}^2, \tag{4}$$
of the integral formula (1) to be integrable. In general an integrand $h$ is integrable (i.e belongs to $L^1$) if and only if
(i) the integrand $h$ is Lebesgue measurable, and
(ii) the absolute value $|h|$ of the integrand has a finite integral $\int |h| dm<\infty$.
Note that $|h(u,v)|=|f(y+u)|~|g(y+v)|$ factorizes in a $u$- and a $v$-dependent factor. Assuming that both functions $f$, $g$ (and therefore $h$) are Lebesgue measurable, this means (via Tonelli's and Fubini's theorems) that the integrand (4) is integrable $h\in L^1(\mathbb{R}^4)$ if and only if (i) both $f,g\in L^1(\mathbb{R}^2)$ are integrable, or (ii) at least one of the two functions $f$ and $g$ vanishes almost everywhere.
For instance, inserting the two non-integrable first-order polynomials $f(y)=y_{\alpha}$ and $g(y)=y_{\beta}$ inside the integral formula (1) is ill-defined as OP also mentions.
On the other hand, the definition (2) does not involve integrals. So we can put $f(y)=y_{\alpha}$ and $g(y)=y_{\beta}$ in eq. (2) to derive the sought-for formula
$$ y_{\alpha} \star y_{\beta}~=~y_{\alpha} y_{\beta}+ i\hbar\epsilon_{\alpha\beta}. \tag{5}$$
References:
M.A. Vasiliev, Unfolded representation for relativistic equations in 2 + 1 anti-de Sitter space, Class. Quant. Grav. 11, (1994) 649. Note that there is an imaginary unit $i$ missing in the published version of formula (1). A preprint version from KEK Preprint Library has the $i$ factor so it seems that the $i$ was lost during the publishing phase.
C. Zachos, Geometrical Evaluation of Star Products, J. Math. Phys. 41 (2000) 5129, arXiv:hep-th/9912238.
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$^1$ Note that Ref. 1 uses the convention
$$ \epsilon_{\alpha\beta} \epsilon^{\beta\gamma}~=~-\delta_{\alpha}^{\gamma}. \tag{6} $$
