First, it would be easier to help if you provided a reprex.
Second, I assume in this dataframe, each row is a sale. Are you looking for products that did not even have a single sale in a region, or is it real-life data and there could always be exceptions? In the first case, you can use table(region, product_id) to get exact numbers, or, for a specific product, something like:
sum(df$product_id == "FUR-BO-10001789" & df$region == "West")
In the second case (real-life, approximate data), I would suggest to start with plotting. Perhaps you could do:
library(ggplot2)
ggplot(df) +
geom_bar(aes(x=product_id, y = ..count.., fill = region),
position = position_stack())
Note that for that approach, if you have enough data, it might be best to first separate your dataframe into an exploration set and a validation set: you can use the first one to make plots and formulate hypotheses, then the second one to check whether these hypotheses are true on independent data. You can do that with something like:
rows_exploratory_subset <- sample(nrow(df), floor(nrow(df)/2))
exploratory_df <- df[rows_exploratory_subset, ]
validation_df <- df[-rows_exploratory_subset, ]