It is truly a very nice question, one of those questions with an answer one feels must be right, but it is not so clear at first how to prove it.
Nevertheless, I claim that there can be no $\vec\forall\vec\exists$ definition of continuity that works in all real-closed fields, meaning a definition with quantifier complexity $\forall x_0\forall x_1\cdots\forall x_n\exists y_0\exists y_1\cdots \exists y_k\varphi(\vec x,\vec y)$.
The basic reason that there can be no $\vec\forall\vec\exists$ characterization of continuity is that continuity is not preserved to limits of chains of models. To see this, let us start with the real field $\langle\newcommand\R{\mathbb{R}}\R,+,\cdot,-,0,1,<\rangle$, and build an elementary tower of hyperreal models over it, each with more infinitesimals with respect to the previous. $$\R\prec\R^*_1\prec \R^*_2\prec\cdots$$ Each of these is a real-closed field with the same theory as the real field, and also for the union of the elementary chain $\R^*_\omega$.
Now, begin with the constant zero function $f_0(x)=0$ in the bottom field (the reals). In each hyperreal field $\R^*_n$, let $f_n$ extend the previous function, still mostly zero, except that we add a new continuous narrow spiking bump from $0$ up to $1$ and back down to $0$ in the new infinitesimal region of $\R_n$ with respect to the previous model.
Thus, each $f_n$ adds one more flashing bump up to $1$ and back down in the new infinitesimal region of $\R_n$, and $f_n$ has $n$ such bumps. All the functions have $f_n(0)=0$.
When we expand the language to include these functions, we get a chain of models. $$\langle \R,+,\cdot,-,0,1,<,f_0\rangle\subseteq \langle \R^*_1,+,\cdot,-,0,1,<,f_1\rangle\subseteq\cdots$$ Let $\R^*_\omega$ be the union of the fields, with the limit function $f$.
Notice that although each $f_n$ was continuous in the $n$th model, nevertheless the limit model does not think the limit function $f$ is continuous, since it has $f(0)=0$ but there are bumps up to $1$ arbitrarily close to $0$. The limit model function is discontinuous.
In short, you can always add one more big bump near zero while staying continuous, but the limit model will not think the limit $f$ is continuous, since it has those jumps up to 1 arbitrarily close to 0.
It follows that the property of continuity is not preserved by unions of chains, and so it cannot be characterized by a $\vec\forall\vec\exists$ property, since such kind of properties always are preserved to limits of chains.
Remark on the underlying theory. The argument shows that there is no $\vec\forall\vec\exists$ definition that works in all real-closed fields. Emil mentions in the comments, however, that we should be using a stronger theory, namely, the theory $T$ that is true in all structures of the form $\langle\R,+,\cdot,-,0,1,<,f\rangle$ for any choice of function $f:\R\to\R$. That is, what we naturally want is a characterization that works in all these structures. The argument I have given does not quite show that there is no $\vec\forall\vec\exists$ definition of continuity in these structures, since perhaps such a definition works in all these models, but not in all real-closed fields. Since my functions $f_n$ can be taken as definable, each of the models in the tower I build can be taken to satisfy the common theory, and although the limit model is a real-closed field, there seems little reason to suppose it satisfies the common theory. (Indeed, one can arrange that it does not, by coding some forbidden information into the limit function, a little at a time.)