You can use the vectorized function if_else() directly:
my_data_inflow$Month <- if_else("7-Oct-19" %in% my_data_inflow$Date, "yes", "no")
As a side-note, depending on the next steps, it may not be ideal to store the result as text "yes"/"no". If you simply write:
my_data_inflow$Month <- "7-Oct-19" %in% my_data_inflow$Date
You get a boolean variable that can easily be reused for other operations.
EDIT: and I forgot to explain the problem: indeed Date is not found, because it is a column of the data frame, you need to indicate that to R with my_data_inflow$Date. It is different if you use the tidyverse function mutate(), perhaps that's what you had in mind:
my_data_inflow <- my_data_inflow %>%
mutate(Month = if_else("7-Oct-19" %in% Date, "yes", "no"))
But that is because mutate() and other dplyr functions use a particular method to allow calling column names with the data frame implicitly declared. This only works inside those functions.