Are $\Pi_2$ consequences of universal theories witnessed by finitely many terms?

I assume that $P$ is supposed to be quantifier-free; otherwise, as discussed in the comments, the result is false.

First, note that since $T$ consists of universal sentences, if $M$ is any model of $T$, then any substructure of $M$ is also a model of $T$. In particular, given $a\in M$, the substructure $N$ generated by $a$ consists of elements represented by terms in $a$, and $N\models\exists y P(a,y)$. Thus, there is a term $t(a)$ such that $N\models P(a,t(a))$, and hence $M\models P(a,t(a))$ as well since $P$ is quantifier-free.

So, for any $a$ in any model of $T$, there is some term $t(a)$ such that $P(a,t(a))$ is true. In other words, adding a constant symbol $c$ to the language, the infinite disjunction $\bigvee_i P(c,t_i(c))$ is true in every model of $T$, where $t_i$ ranges over all possible terms. By compactness, there must be finitely many $t_1,\dots,t_n$ such that $T\models\bigvee_{k=1}^n P(c,t_k(c))$, and so $T\models \forall x \bigvee_{k=1}^n P(x,t_k(x))$