I missed the update to your post where you ask about weights and small sample sizes–my apologies for not responding to your entire query.
Weights will inflate your counts to the estimated “true” population size. That being said, depending on your unweighted sample size, it may not be appropriate to make population inferences from such a small group. In general, there is no bright-line rule regarding “too small” to study. Although I can say that more is always better, and one observation is certainly too small. In practice, what will happen is the sampling error around estimated statistics will be relatively large and will, therefore, limit any informative interpretation from the data.
One way to increase the sample size of your estimates is to pool multiple samples together (e.g., across various months to combine multiple ORGs). This will increase the number of observations in your data and the statistical precision but will limit the temporal precision of your analysis. Note that if you do pool together multiple samples you will need to adjust the sampling weights so that they properly account for the combined samples. An approximate way to do this is to divide the sampling weight by the number of samples you are pooling together.