I recommend against using a continuous distribution to approximate a discrete distribution. Often it will work well enough, especially in data like yours where the counts are far from zero, but the further into the tails you get the less accurate it will be, so you have to be vigilant to make sure that you are not pushing the approximation too far. The appropriate discrete distributions are not any harder to use, so it's better just to use them.
The Poisson Distribution
For count data where the rate is constant, the distribution you want is the Poisson Distribution. You can think of the Poisson distribution (and in fact derive its mathematical form) as a limiting case of the binomial distribution with $N \rightarrow \infty$ and $p \rightarrow 0$, such that the expected number of successes $Np$ is constant. The Poisson distribution has support for counts $\ge 0$, which is important because for low rates you will sometimes see zero counts, but of course you will never see a negative number of counts. A normal distribution cannot be restricted to be positive, while a lognormal distribution cannot produce zero counts, so neither one of these is a good choice for low average event rates.
The Gamma-Poisson (aka Negative Binomial) Distribution
Because the Poisson distribution has only one parameter, its variance and mean are not independently selectable. In fact, if the parameter in the distribution is denoted by $\lambda$, then the variance and mean are both equal to $\lambda$. This may not be appropriate if your data shows evidence of having a variance that is larger than the mean, a feature called "overdispersion". In this case, you want the Gamma-Poisson Distribution. The Gamma-Poisson distribution is a mixture of Poisson distributions, where the parameter $\lambda$, instead of being constant like in the regular Poisson distribution, has a gamma distribution. This has the effect of smearing out the distribution a bit, so as to increase its variance. The width of the gamma distribution controls the amount of smearing, allowing you to choose the amount of overdispersion that's appropriate for your data.
As it happens, the Gamma-Poisson distribution is equivalent to the Negative Binomial Distribution, and that's the name you will find it under in most stats software packages. The parameterization for the negative binomial is a little confusing for this application, but you can sidestep that by using the mean and variance to choose the parameters.
The Conway-Maxwell-Poisson Distribution
The Conway-Maxwell-Poisson (CMP) Distribution generalizes the Poisson Distribution by adding an additional parameter that accounts for either overdispersion or underdispersion. Compared to the Gamma-Poisson, this distribution has the advantage of being able to handle underdispersed as well as overdispersed data (the Gamma-Poisson can only handle overdispersed). However, this flexibility comes at the cost of being a little more obscure and less well supported than the distributions above. The COMPoissonReg R package implements the CMP distribution, as well as CMP regressions using generalized linear models (GLMs)
Another way of handling underdispersion
Often underdispersion results from processes that have a component that isn't random. For example, you might have a customer whose order comes in reliably every week for the same amount of product, and added to this you might have irregular orders that display more variability. In this case, you can model the system as a constant distribution plus a Poisson distribution. For example, if you average 100 counts per week, but the variance is only 64, then you could model this as a flat 36 counts, plus a Poisson distribution with $\lambda = 64$.
As noted in the comments, however, the variance of a small sample is highly uncertain, so before introducing a correction for underdispersion, you should consider whether you have reason to expect this sort of behavior.
