As an opening comment, which you should not use to try to influence your management (especially given that you are an intern), Type I service level is almost always a terrible objective. What it measures is the probability that you suffer no stockouts over the leadtime, but, from a business perspective, what you'd like to measure is, far more often, the expected number of stockouts over the order cycle $/$ the expected demand over the order cycle, i.e., the fraction of demands that go unfilled. In some situations, e.g. stocking a helicopter for rescue missions where an out of stock could mean permanent disability or death, Type I service level is appropriate, but not in the usual business case. I won't go into more detail, since it's off-topic, but include it here for your future information.
The Type I service level given an order point of $s$ is nothing more than the probability of seeing fewer than $s$ demands over leadtime. As such, if you have a probability distribution of leadtime demand $F(x)$, and a target service level $\alpha$, the corresponding order point $s_{\alpha}$ is:
$$s_{\alpha} = F^{-1}_{LTD}(\alpha)$$
where $LTD$ refers to "leadtime demand". In the case of the Normal distribution, it so happens that this is the same as:
$$s_{\alpha} = \mu_{LTD} + \sigma_{LTD}*z_{\alpha}$$
where $\mu$ is the mean demand, and $\sigma$ is the standard deviation of demand. Where does $z_{\alpha}$ come from? It's the inverse of the standard Normal distribution's cumulative density function at $\alpha$:
$$z_{\alpha} = F^{-1}(\alpha | \mu=0, \sigma=1)$$
and
$$F^{-1}(\alpha | \mu_{LTD}, \sigma_{LTD}) = \mu_{LTD} + \sigma_{LTD} F^{-1}(\alpha|\mu=0,\sigma=1) $$
So using $z_{\alpha}$ with the Normal distribution is the same as calculating the inverse of the cumulative distribution of leadtime demand, where leadtime demand is Normally distributed.
When you aren't working with the Normal distribution, you won't typically have any equivalent formula to make life appear simpler. This is true of both the Poisson and Negative Binomial distributions. In the case of these two distributions, it's most straightforward just to calculate order point from the initial equation, using the appropriate parameters for the leadtime demand distribution in question.
ETA:
For example, assume your daily demand has mean $0.2$ units and standard deviation $0.6$ units, and you have a leadtime of one week. Then $\mu_{LTD} = 1.4$ and $\sigma_{LTD} = 1.59$. The parameters of the Negative Binomial distribution are $r = 1.75$ and $p = 0.556$.
If you want to stock to a Type I service level of $95\%$, you'd solve for:
$$s_{\alpha} = F^{-1}_{LTD}(\alpha) = F^{-1}(\alpha=0.95;r=1.75,p=0.556) = 5$$
and your order point would be $5$.
Best Answer
There are many power-law distributions, so you have a lot of possible models. You might start by trying to fit a log-series distribution, which is a limiting case of the negative binomial.
Don't think a priori that you have a mixture distribution as suggested by whuber until you've estimated model parameters and done at least a goodness of fit test. Long-tail distributions, like power-law, log-series, Zipf, etc., typically have what look like outliers in the right-hand tail; their separation from the bulk of the observations is just an artifact of (relatively) small sample size. Mixtures are a pain in the butt to estimate, since some regions overlap. You can often avoid that sort of problem by stepping up your modeling one level with something like Poisson regression, assuming you have some covariate (predictor) data about each user -- this basically does the mixing for you.
The Johnson, Kemp, and Kotz reference given at the end of the referenced Wikipedia article has everything you'd ever want to know about all these distributions, including many methods of parameter estimation.