Is there an equation for the median and percentile of a zero-inflated Poisson? This question gives answer to the mean and variance of the zero inflated Poisson distribution, but is there an equation for the median or the percentile?
 A: Somewhat surprisingly, it seems like we don't have anything on quantiles for zero-inflated distributions, so I'll answer the more general question.
Assume you have a zero-inflated discrete distribution, i.e., your observation is zero with probability $\pi$ and comes from some other specified "vanilla" discrete distribution with probability $1-\pi$, so the probability mass function is
$$ f_\pi(y) = 
\begin{cases}
\pi+(1-\pi)f_0(0), & \text{if }y=0 \\
(1-\pi)f_0(y), & \text{if }y=1,2....
\end{cases}
$$
where $f_0$ denotes the PMF of the discrete "vanilla" distribution.
Consider a percentage $0\leq p<1$. Then the $p$-th quantile of the zero-inflated distribution is
$$q_\pi(p) = 
\begin{cases}
0, & \text{if } p\leq \pi+(1-\pi)f_0(0) \\
q_0\big(\frac{p-\pi}{1-\pi}\big), & \text{if } p> \pi+(1-\pi)f_0(0),
\end{cases}
$$
where $q_0(p)$ denotes the $p$-th quantile of the "vanilla" distribution.

In your specific case, the "vanilla" distribution is a Poisson distribution with parameter $\lambda$. There does not seem to be a closed form for the quantile of a "vanilla" Poisson distribution (R's ?qpois uses a search), so I would not expect there to be one for the more complicated zero-inflated case, either.

To see this, note first that if your target percentage $p$ fulfills $p\leq \pi+(1-\pi)f_0(0)$, then your percentile $q_\pi(p)$ is obviously zero.
So let us assume that $p>\pi+(1-\pi)f_0(0)$. We just follow the definition of a quantile:
$$ \begin{align}
q_\pi(p)
= & \min\Big\{y\,|\,\sum_{x=0}^y f_\pi(y)\geq p\Big\} \\
= & \min\Big\{y\,|\,\pi+(1-\pi)f_0(0)+
(1-\pi)\sum_{x=1}^yf_0(x)\geq p\Big\} \\
= & \min\Big\{y\,|\,\pi+(1-\pi)\sum_{x=0}^yf_0(x)\geq p\Big\} \\
= & \min\Big\{y\,|\,\sum_{x=0}^yf_0(x)\geq \frac{p-\pi}{1-\pi}\Big\} \\
= & q_0\big(\frac{p-\pi}{1-\pi}\big),
\end{align}$$
as claimed.
