Proving moduli of uniform continuity in RCA_0 Simpson's Subsystems of Second Order Arithmetic (pp. 134ff.) uses RCA$_0$ to prove various theorems of analysis for  all continuous functions with a suitable modulus of uniform continuity.  And he notes that RCA$_0$ itself proves "any continuous function which arises in practice" does have such a modulus.  Is there a place I can go to see many examples of the functions this covers?  I am especially interested in various complex analytic functions arising in analytic number theory. 
 A: On a space like $\mathbb{R}^n$, $[0,1]^n$, or $\mathbb{C}$, every uniformly continuous function that "arises in practice" (more on this in a bit) has a computable modulus of continuity.
To find examples, just open up a book on your favorite subject (in this case complex analysis).  Every particular uniformly continuous function in that book will likely have a computable modulus of uniform continuity.  To find this modulus of continuity, just follow that proof that your particular function is uniformly continuous.
For example, $e^x$ is uniformly continuous on $[0,1]$.  Just use that $e^x$ has a computable Taylor series and the error estimate with that Taylor series is computable and uniform over $[0,1]$.  The same goes for other functions like $\sin(x)$.
It might help to also know that if $f\colon K \rightarrow X$ is a computable function on a computable complete compact metric space $K$ to a computable Polish space $X$, then $f$ also has a computable uniform modulus of continuity.  (To see this, follow the proof the every continuous function $f\colon K \rightarrow X$ is uniformly continuous.  I imagine the details can be found in Weihrauch's book---but I haven't checked.)

Additional Remarks:
Let's call this principle the "continuity thesis": 

Every uniformly continuous function that "arises in practice"  has a computable modulus of continuity

The continuity thesis is not something that I know how to state formally, much less prove.$^*$  Nonetheless, it seems to be a deep principle in mathematics, similar to the Church-Turing thesis.
I basically take the continuity thesis to be a given whenever I do research in the area.  Nonetheless, just like the Church-Turing thesis, it is very instructive to go through a large number of examples to convince yourself why it is true.
As for why the continuity thesis holds, it seems to be that the only way one can prove that a function on, say, $[0,1]$ is continuous is to explicitly give it's modulus of uniform continuity (or a similar quantitative witness to uniform continuity).
Last, I am sure many mathematicians wouldn't say this principle is as strong as the Church-Turing thesis, since "arises in practice" is quite vague.  Nonetheless, I challenge them to find a counterexample in an analysis book.

$^*$ Concerning the provability of the continuity principle, I think you (slightly) misrepresented Simpson's quote (pp.136-137 in the second edition):

However, it is interesting to note that “any continuous function which arises in practice” can be proved in $\mathsf{RCA}_0$ to have a modulus of uniform continuity on any closed bounded subset of its domain.

