I had to add a prefix of 01 to the dates to convert them with lubridate.
I do not know how to format the date within the data frame and I cannot think of a reason to do it. If you want to format the character representation of the date, you can use the format function.
library(lubridate)
DF <- data.frame(Year = c("Jan-14", "Feb-14"), stringsAsFactors = FALSE)
DF$FullDate <- paste0("01-", DF$Year)
DF$FullDate <- dmy(DF$FullDate)
DF
#> Year FullDate
#> 1 Jan-14 2014-01-01
#> 2 Feb-14 2014-02-01
format(DF$FullDate, "%d/%m/%Y")
#> [1] "01/01/2014" "01/02/2014"
Created on 2020-06-08 by the reprex package (v0.3.0)