Just to summarise as an answer the line of thought in the comments: determining whether a function is periodic requires looking at an infinite number of its coefficients, since any finite Taylor series can be the initial segment of a periodic function, and indeed a periodic function with arbitrary period. So there is no decision algorithm for the question if the Taylor series is provided by an oracle giving one coefficient at a time.
On the other hand if you can evaluate $f(q)$ at every rational number, you can find all members of the discrete countable set of zeros of $f$ in $\mathbb{R}$; then testing whether $f(x-z) - f(x) \equiv 0$ at each such zero $z$ gives you an infinitary method relying only on the most obviously natural operations on the whole series $(a_n)$ and making a countable number, (as an ordinal, $2 \cdot \omega$) of decisions, to come to an answer.
Clearly this cannot count as "reasonably direct" because that would be a deeply uninteresting answer to an interesting question. So to make the question precise we would have to frame it in a way that lies between these two cases. What that should be, I have no idea.