Often, the Independence Property is defined in the monster model of a complete theory. When it is not, it usually goes like: a formula $\phi(x, y)$ is said to have the independence property if for every model $A$ of the theory, and for every $n \in \omega$, there is a family of tuples $b_0, \ldots, b_{n-1}$ s.t. for every subset $X$ of $[n]$ there is a tuple $a \in A$ for which
$$A \vDash \phi(a, b_i) \leftrightarrow i \in X.$$
My question is: can the hypothesis for every model of the theory be replaced by there is a model of the theory?
Yes, because the property is elementary. That is, if our theory $T$ is complete and consistent, and in some model $\varphi(x,y)$ has the independence property, then for each $n\in \omega$, the following sentence is in the theory: $$\exists y_0,\dots,y_{n-1}\, \bigwedge_{X\subseteq [n]} \left(\exists x \bigwedge_{i \in X} \varphi(x,y_i)\land \bigwedge_{i \notin X} \lnot\varphi(x,y_i)\right),$$ so it is satisfied in all models.