We can prove the approximate intermediate value theorem constructively using only pointwise continuity. The proof has the same feel as $\forall x \in \mathbf{R}\ \exists n \in \mathbf{N} \ n > r$, which is constructively valid but with $n$ chosen in a way that depends on the particular rational sequence defining $x$.
A real number $x$ is defined to be a sequence of rationals $x^n$ such that $|x^m-x^n|\le 1/m+1/n$. (Since there are no positive exponents in this proof, all positive superscripts will be these rational approximations.) So $|x-x^n| \le 1/n$ and, e.g. we can choose the $n$ above to be $\lceil x^1 \rceil + 2.$
Now we are given $a,\ b,\ \epsilon,\ f$ as in the question. Let $a_1=a$, $b_1=b$.
$$\text{Let }c_n = (a_n+b_n)/2.$$ $$\text{If }f(c_n)^n < 0,\text{ then let }a_{n+1} = c_n,\ b_{n+1}=b_n.$$ $$\text{If }f(c_n)^n \ge 0,\text{ then let }a_{n+1} = a_n,\ b_{n+1}=c_n.$$
Unlike the version referenced in the 11/16 comment, this is deterministic at each stage, so the construction of the $c$'s requires only unique choice and not dependent choice. (If there's a hidden use of dependent choice, please let me know!)
The intervals $[a_n,b_n]$ have lengths decreasing by halves with intersection $c$. Furthermore, $f(a_n)^n < 1/n$ and $f(b_n)^n \ge 1/n$ for all $n$.
By pointwise continuity of $f$, choose $\delta$ such that $|x-c|<\delta$ implies $|f(x)-f(c)|<\epsilon/3$.
Choose $m$ with $(a-b)2^{-m} < \delta$ and $1/m < \epsilon/3$. Then
$$|a_m-c|<(a-b)2^{-m} < \delta, \ \text{ and }\ f(c) \le f(a_m) + \epsilon/3 \le f(a_m)^m + 1/m + \epsilon/3 \le \epsilon.$$ By similar comparison with the $b$'s, $f(c) \ge -\epsilon$. So $c$ is as desired to prove the approximate intermediate value theorem, QED.