Polynomial distributions on $\mathbb P^n$. The following works for any field $k$.
The polynomial $1$-forms defined on $\mathbb A^{n+1}$ which induce distributions on $\mathbb P^n$ are those invariant by homotheties and annihilated by the Euler vector field $R = \sum_{i=0}^n x_i \partial_i$. Explictly these can be written as
$$
\omega = \sum_{i=0}^n A_i dx_i
$$
with $A_0, \ldots, A_n$ being homogeneous polynomials of degree $d+1$ satisfying the relation
$$
\sum_{i=0}^n x_i A_i =0 .
$$
In more intrinsic terms $\omega$ is section of $\Omega^1_{\mathbb P^n}(d+2)$. The integer $d$ appearing above has a nice geometric interpretation when $k=\overline k$ is an algebraically closed field. If we consider a linear inclusion $i: \mathbb P^1 \to \mathbb P^n$ then
$i^* \omega$ is a section of $\Omega^1_{\mathbb P^1}(d+2) \simeq \mathcal O_{\mathbb P^1}(d)$ and therefore $d$ counts the number of tangencies between the distribution defined by $\omega$ with a generic line. We say that $d$ is the degree of the distribution.
Be careful: the degree of a distribution on $\mathbb P^n$ as defined above does not
coincide with the degree of the coefficients of a polynomial $1$-form defining the same distribution in affine coordinates. Indeed the (maximal) degree of the affine polynomials defining the distribution on $\mathbb A^{n}$ is equal to $d+1$.
Examples of polynomial contact structures on $\mathbb R\mathbb P^3$ of even degree. The contact
structures on $\mathbb R^3$ defined by
$$
(qy−rz+a)dx+(pz−qx+b)dy+(rx−py+c)dz ,
$$
with $ ap+br+cq \neq 0 $, all have degree zero as they can be written in homogenous coordinates $(x:y:z:w) \in \mathbb P^3$ as
$$
(qy−rz+aw)dx+(pz−qx+bw)dy+(rx−py+cw)dz + (-ax -by - cz ) dw .
$$
It can also be checked that the induced distributions are all on the $PGL(4,\mathbb R)$-orbit of the one defined by
$$
\omega_0 = xdy- ydx + zdw- w dz .
$$
Indeed, the action of $\mathrm{PGL}(4,\mathbb C)$ on $\mathbb P H^0 ( \mathbb P^3, \Omega_{\mathbb P^3}(2))$ has only two orbits. The closed one corresponds to the integrable $1$-forms ( foliations singular along a line )
while the open one corresponds to contact structures.
Clarification. The space $\mathbb PH^0(\mathbb P^3, \Omega^1(2))$ can be naturally identified with $\mathbb P ( \bigwedge^2 \mathbb C^4)$. Indeed, the exterior differential is an injective map from linear homogeneous $1$-forms annihilated by Euler's vector field to constant $2$-forms; and the interior product with Euler's vector field sends constant
$2$-forms to linear homogeneous $1$-forms annihilated by Euler's vector field. Under these
maps the integrable $1$-forms correspond to decomposable $2$-forms. In other words, the foliations in $\mathbb P H^0(\mathbb P^3, \Omega^1(2))$ correspond to the Plucker embedding of the Grasmannian of lines in $\mathbb P^3$ into $\mathbb P (\bigwedge^2 \mathbb C^4)$.
To produce polynomial contact structures of any even degree $2d$ we have just to multiply $\omega_0$
by an even homogenous polynomial $P_{2d} \in \mathbb R[x,y,z,w]$ without non-trivial real solutions and perturb the result in $H^0(\mathbb R \mathbb P^3, \Omega^1(2d+2))$. Since
$$
(P_{2d} \omega_0) \wedge d (P_{2d} \omega_0) = P_{2d}^2 \omega_0 \wedge d \omega_0
$$
does not vanish at any point of $\mathbb R \mathbb P^3$,
we obtain that any section of $ \Omega^1(2d+2) $ in a Zariski sufficiently small
(analytic) neighborhood of $P_{2d}\omega_0$ also defines a contact structure.
There are no polynomial contact structures of odd degree on $\mathbb R \mathbb P^3$.
If we have a nowhere zero section of real vector bundle $E$ on a compact manifold $X$ then
the top Stiefel-Whitney class of $E$ vanishes. From Euler's sequence
$$
0 \to \Omega^1_{\mathbb R \mathbb P^n} \to \mathcal O_{\mathbb R \mathbb P^n}(-1)^{\oplus n+1} \to \mathcal O_{\mathbb R \mathbb P^n} \to 0
$$
we can deduce that
$$
w_n( \Omega^1_{\mathbb R \mathbb P^n}(d+2) ) = \sum_{i=0}^n (-1)^i (d+1)^{n-i} \mod 2 .
$$
Notice that the same formula (without the $\mod 2$) counts the number of singularities of
a polynomial distribution over an algebraically closed field if the singularities are isolated.
Specializing to $\mathbb R \mathbb P^3$ we get
$$
w_3 ( \Omega^1_{\mathbb R \mathbb P^3}(d+2) ) = \left\lbrace
\begin{array}
00 &\text{ if } d \text{ is even} \newline
1 &\text{ if } d \text{ is odd}
\end{array}\right.
$$
and we see that there are no contact distributions of odd degree on $\mathbb R \mathbb P ^3$.
Historical remark. The inexistence result above can be traced back to Habicht (1948). He dealt with a somewhat different problem which admits an equivalent algebraic formulation. His motivation came from Poincaré-Brower
Theorem about the inexistence of continuous vector fields on the sphere $S^2$. If one looks for homogeneous polynomial vector fields on $\mathbb R^{n+1}$ tangent to the unitary sphere $S^n$ one ends
up with $n+1$ homogeneous polynomials $(f_0, \ldots, f_n)$ satisfying $\sum x_i f_i=0$. Of course, this is the same as homogeneous polynomial $1$-forms annihilated by Euler's vector field.
