Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a smooth function with a single root we call $y_*$. Then define
\begin{equation} F_{\delta}:=\int^{y_*+ \delta/2}_{y_*- \delta/2} 1/f(y)dy \end{equation}
Let $\delta \ll 1$. Is it possible to write $F_{\delta}$ as a taylor series? By which I mean \begin{equation} A + B \delta + C \delta^2 + \cdots \end{equation} If so, how do I arrive at the expressions for $B,C$ etc. ?
It depends on the type of singularity. Let's say that $f(y) \sim (y-y_*)^{a}$ near this singularity. If $a \lt 1$ then the singularity is integrable and you'll get something like
$$\frac{1}{1-a} \left [(y-y_*)^{1-a}\right]_{y_*-\delta/2}^{y_*+\delta/2}$$
which you will find has some complex part. If $a \ge 1$, on the other hand, the integral diverges and your proposed expansion is invalid.
You may have something in mind like a Cauchy principal value, in which you consider the limit as $\delta \to 0$, but I am not aware of any meaningful expansion in delta.