Setting aside the assumption that $\phi$ be a polynomial mapping for the moment, if one makes the 'nondegeneracy' assumptions
1. $\mathrm{dim}\ \mathrm{span}\bigl( \phi_s(s,t), \phi_t(s,t)\bigr) =2 $,
2. $\mathrm{dim}\ \mathrm{span}\bigl( \phi_s(s,t), \phi_t(s,t), \phi_{ss}(s,t), \phi_{st}(s,t), \phi_{tt}(s,t) \bigr) = 3$ for all $(s,t)\in[0,1]^2$, and
3. the subspace $W(s,t) =\mathrm{span}\bigl( \phi_s(s,t), \phi_t(s,t), \phi_{ss}(s,t), \phi_{st}(s,t), \phi_{tt}(s,t) \bigr)\subset\mathbb{R}^4$ is not constant, in the sense that $W:[0,1]^2\to \mathrm{Gr}_3(\mathbb{R}^4)\simeq\mathbb{RP}^3$ has nonvanishing differential,
then one can show that the surface $\phi\bigl([0,1]^2\bigr)\subset\mathbb{R}^4$ is ruled (and does not lie in an affine $3$-space).
Such surfaces locally depend on three arbitrary functions of one variable in Cartan's sense. One way of describing them is this: Let $\Lambda$ be the space of (affine) lines in $\mathbb{R}^4$, a manifold of dimension $6$. Consider the ($9$-dimensional) bundle $\pi:F\to\Lambda$ whose fiber over $\lambda\in\Lambda$ is the flag variety of the $3$-dimensional vector space $\mathbb{R}^4/\lambda'$, where $\lambda'\subset\mathbb{R}^4$ is the linear subspace parallel to $\lambda$. Then there exists a smooth $4$-plane field $D\subset TF$ such that, if $\gamma\subset F$ is a generic curve tangent to $D$, then regarding $\gamma$ as a $1$-parameter family of affine lines via $\pi(\gamma)\subset \Lambda$, the union of these lines is a surface satisfying the above assumptions. (Here 'generic' means that the tangents to $\gamma$ do not lie in triple of hyperplanes in $D$.)
I don't know, offhand, whether there are any polynomial solutions to these conditions, but I think that shouldn't be hard to answer. (I suspect that there are such polynomial solutions, but I haven't had time to check this.)
The following analysis is a more-or-less standard approach using
the so-called 'moving frame'. (I'm sure that the result itself
is classical in some sense, though I don't know offhand where to look
in the literature to find it.)
The conditions
1. $\mathrm{dim}\ \mathrm{span}\bigl( \phi_s(s,t), \phi_t(s,t)\bigr) =2 $,<p>
2. $\mathrm{dim}\ \mathrm{span}\bigl( \phi_s(s,t), \phi_t(s,t), \phi_{ss}(s,t), \phi_{st}(s,t), \phi_{tt}(s,t) \bigr) = 3$ for all $(s,t)\in[0,1]^2$, and <p>
3. the subspace $W(s,t) =\mathrm{span}\bigl( \phi_s(s,t), \phi_t(s,t), \phi_{ss}(s,t), \phi_{st}(s,t), \phi_{tt}(s,t) \bigr)\subset\mathbb{R}^4$ is not constant, in the sense that $W:[0,1]^2\to \mathrm{Gr}_3(\mathbb{R}^4)\simeq\mathbb{RP}^3$ has nonvanishing differential,<p>
are actually independent of the choice of $st$-coordinates on the surface in $\mathbb{R}^4$ and so can be regarded as conditions on a surface $S\subset\mathbb{R}^4$ that, for local analysis purposes, can be taken to be smoothly embedded.<p>
Let $B_0(S)$ denote the space of quintuples $(p;v_1,v_2,v_3,v_4)$ where $p$ lies in $S$ and the quadruple $(v_1,v_2,v_3,v_4)$ is a basis of $\mathbb{R}^4$ such that $(v_1,v_2)$ is a basis of $T_pS$ while $(v_1,v_2,v_3)$ is a basis of the $3$-dimensional subspace $W_pS\subset \mathbb{R}^4$. Then $B_0$ is a smooth submanifold of the product $\mathbb{R}^4\times\mathrm{GL}(4,\mathbb{R})$ that has dimension $2 + 4 + 3 + 4 = 13$.
<p>
It is useful to define $\mathbb{R}^4$-valued functions $x,e_i: \mathbb{R}^4\times\mathrm{GL}(4,\mathbb{R})\to\mathbb{R}^4$ such that $x(p;v_1,v_2,v_3,v_4) = p$ while $e_i(p;v_1,v_2,v_3,v_4) = v_i$. Then there
exist unique (linearly independent) $1$-forms $\omega^i$ and $\theta^i_j$
on $\mathbb{R}^4\times\mathrm{GL}(4,\mathbb{R})$ such that (assuming the
usual summation convention on repeated indices) the following *structure equations* hold:
$$
\mathrm{d}x = e_i \omega^i
\qquad\text{and}\qquad
\mathrm{d}e_i = e_j\,\theta^j_i\,.\tag1
$$
$$
\mathrm{d}\omega^i = -\theta^i_j\wedge\omega^j
\qquad\text{and}\qquad
\mathrm{d}\theta^i_j = -\theta^i_k\wedge\theta^k_j\,.\tag2
$$
Now, let us pull back these functions and $1$-forms to $B_0(S)$ (but, as is customary, not notate the pullback). The definition of $B_0(S)$ implies that
$$
\mathrm{d}x\wedge e_1\wedge e_2 = \mathrm{d}e_1\wedge e_1\wedge e_2\wedge e_3
= \mathrm{d}e_2\wedge e_1\wedge e_2\wedge e_3 = 0
$$
while the two expressions
$$
\bigl(\,\mathrm{d}e_1\wedge e_1\wedge e_2,\ \mathrm{d}e_2\wedge e_1\wedge e_2\bigr)
\qquad\text{and}\qquad
\mathrm{d}e_3\wedge e_1\wedge e_2\wedge e_3
$$
are nowhere vanishing. Using the above structure equations, these are equivalent to the relations
$$
\omega^3 = \omega^4 = \theta^4_1 = \theta^4_2 = 0,\tag3
$$
while $\omega^1\wedge\omega^2$ is nonvanishing, $\theta^4_3$ is nonvanishing, and the pair $(\theta^3_1,\theta^3_2)$ do not simultaneously vanish.<p>
Meanwhile, using the structure equations, we have
$$
0 = \mathrm{d}\omega^3
= -\theta^3_1\wedge\omega^1 -\theta^3_2\wedge\omega^2,
$$
so there must exist functions $h_{ij}=h_{ji}$ for $1\le i,j,\le 2$, not all simultaneously vanishing, such that $\theta^3_i = h_{ij}\omega^j$. The quadratic form $h = h_{ij}\,\omega^i\omega^j$ is then nonvanishing and well-defined up to multiples on the surface $S$. Moreover, for $1\le i\le 2$, we have
$$
0 = \mathrm{d}\theta^4_i
= -\theta^4_k\wedge\theta^k_i = -\theta^4_3\wedge\theta^3_i\,.
$$
Thus, since $\theta^4_3$ is nonvanishing, it follows that $\theta^3_1$ and $\theta^3_2$ are multiples of $\theta^4_3$. In particular, $\theta^3_1\wedge\theta^3_2$ vanishes identically, so $h_{11}h_{22}-{h_{12}}^2$ vanishes identically. Thus, the quadratic form $h$ has rank $1$.
Let $B_1(S)\subset B_2(S)$ denote the submanifold defined by $h_{11} = h_{12}=0$. It is a smooth submanifold of $B_0(S)$ of dimension $11$, and when all the forms and functions are pulled back to $B_1(S)$, we have $\theta^3_1 = 0$ while $\theta^3_2 = h_{22}\,\omega^2$. In particular, it now follows that $\theta^4_3$ is also a multiple of $\omega^2$, say $\theta^4_3 = f\,\omega^2$ for some $f$ (which is nonvanishing). <p>
Further, we have
$$
0 = \mathrm{d}\theta^3_1 = -\theta^3_k\wedge\theta^k_1
= -\theta^3_2\wedge \theta^2_1 = -h_{22}\omega^2\wedge\theta^2_1,
$$
so it follows (since $h_{22}$ is nonvanishing) that $\theta^2_1 = g\,\omega^2$ for some function $g$ on $B_1(S)$.
Now, we can compute that
$$
\mathrm{d}\omega^2 = -\theta^2_j\wedge\omega^j
= -g\,\omega^2\wedge\omega^1 - \theta^2_2\wedge\omega^2
= -(\theta^2_2 - g\,\omega^1)\wedge\omega^2.
$$
Thus, $\omega^2$ is an integrable $1$-form, and, because it is semi-basic
for the submersion $x:B_1(S)\to S\subset\mathbb{R}^4$, it follows that $\omega^2$ is a multiple of the $x$-pullback of a (nonvanishing) $1$-form on $S$. Thus, $S$ is foliated by (connected) curves whose $x$-preimages in $B_1(S)$ are codimension $1$ integral submanifolds of $S$. <p>
I claim that these curves in $S$ are, in fact lines in $\mathbb{R}^4$. To see this, note that, when one pulls back to a leaf of $\omega^2=0$ in $B_1(S)$, one has $\theta^2_1 = \theta^3_1=\theta^4_1= 0$ as well, so one has
$$
\mathrm{d}x = e_1\,\omega^1\qquad\text{and}\qquad \mathrm{d}e_1 = e_1\,\theta^1_1\,.
$$
In particular, the direction of $e_1$ is fixed on this leaf and it is the tangent direction of the mapping $x$ restricted to this leaf. Hence the $x$-image of this leaf is an open interval in a line in $\mathbb{R}^4$. Thus, the surface is ruled, as claimed.<p>
There still remains the question of how one could 'generate' these surfaces, at least locally. I claim that, since these surfaces are, in fact, $1$-parameter families of lines, one should think of them as curves in the space of lines that satisfy some differential equations. Here is how one can think of this system of equations:<p>
Let $\mathcal{I}$ denote the Pfaffian system of rank $5$ that is generated by the five linearly independent $1$-forms
$$
\omega^3,\ \omega^4,\ \theta^4_1\,,\ \theta^4_2\,,\ \theta^3_1
$$
(these are the $1$-forms that vanish when pulled back to $B_1(S)$ when $S\subset\mathbb{R}^4$ is a surface satisfying our hypotheses). By the structure equations,
$$
\left.
\begin{aligned}
\mathrm{d}\omega^4 &\equiv 0\\
\mathrm{d}\theta^4_1 &\equiv 0\\
\mathrm{d}\omega^3 &\equiv -\theta^3_2\wedge\omega^2\\
\mathrm{d}\theta^3_1 &\equiv -\theta^3_2\wedge\theta^2_1\\
\mathrm{d}\theta^4_2 &\equiv \phantom{-}\theta^3_2\wedge\theta^4_3\\
\end{aligned}\ \right\} \mathrm{modulo}\ \mathcal{I}
$$
It follows that $\mathcal{J}$, the Cartan system of $\mathcal{I}$,
has rank $9$ and is spanned by the nine $1$-forms
$$
\omega^3,\ \omega^4,\ \theta^4_1\,,\ \theta^4_2\,,\ \theta^3_1\,,\
\omega^2,\ \theta^2_1\,,\ \theta^3_2\,,\ \theta^4_3
$$
In fact, $\mathcal{J}$ is easily seen to be the bundle of $1$-forms
on $\mathbb{R}^4\times\mathrm{GL}(4,\mathbb{R})$ that is semibasic for the projection to the $9$-manifold $F$, where the projection sends $(p; v_i)$
to the element of $F$ described by
$$
\bigl (\lambda(p,v_1),\ [v_1\wedge v_2], [v_1\wedge v_2\wedge v_3]\ \bigr)
$$
and where $\lambda(p,v_1)$ is the line through $p$ in the direction $v_1$.
In particular $\mathcal{I}$ is the pullback of a well-defined Pfaffian system of rank $5$ on $F$, whose annihilator is the $4$-plane field $D\subset TF$. Moreover, it is easy to show that the three rank $6$ Pfaffian systems got by adjoining any one of $\omega^2$, $\theta^3_2$, or $\theta^4_3$ to $\mathcal{I}$ are themselves pullbacks of rank $6$ Pfaffian systems on $F$ whose annihilators in $TF$ are $3$-plane subbundles of $D$. <p>
By its very construction, the projection of $B_1(S)$ into $F$ is a curve that is tangent to $D$ and not tangent to any of these three $3$-plane subbundles of $D$. Conversely, if $\gamma\subset F$ is any integral curve of $D$ that is not tangent to any of the three $3$-plane subbundles of $D$, its preimage in $\mathbb{R}^4\times\mathrm{GL}(4,\mathbb{R})$ is a submanifold of the form $B_1(S)$ for a surface $S$ satisfying our conditions, in fact, the surface swept out by the union of the lines represented by $\lambda(\gamma)$, where $\lambda:F\to \Lambda$ is the obvious map to the lines.
It is a standard fact that the curves tangent to a 4-plane field in a smooth manifold are locally described (up to reparametrization) by prescribing $3$ functions of one variable.