group by and scale/normalize a column in r

The issue is that you are using the wrong dplyr verb. Summarize will create one result per group per variable. What you want is mutate. Mutate changes variables and returns a result of the same length as the original. See http://cran.rstudio.com/web/packages/dplyr/vignettes/dplyr.html. Below two approaches using dplyr:

df %>%
    group_by(Store) %>%
    mutate(Temperature = normalit(Temperature), Sum_Sales = normalit(Sum_Sales))

df %>%
    group_by(Store) %>%
    mutate_each(funs(normalit), Temperature, Sum_Sales)

Note: The Store variable is different between your data and desired result. I assumed that @jlhoward got the right data.

Here's a data.table solution. I changed your example a bit to have two type of store.

df <- read.table(header=T,text="Store Temperature Unemployment Sum_Sales
1     1       42.31        8.106   1643691
2     1       38.51        8.106   1641957
3     1       39.93        8.106   1611968
4     2       46.63        8.106   1409728
5     2       46.50        8.106   1554807
6     2       57.79        8.106   1439542")

library(data.table)
DT <- as.data.table(df)
DT[,list(Temperature=normalit(Temperature),Sum_Sales=normalit(Sum_Sales)),
    by=list(Store,Unemployment)]
#    Store Unemployment Temperature Sum_Sales
# 1:     1        8.106  1.00000000 1.0000000
# 2:     1        8.106  0.00000000 0.9453393
# 3:     1        8.106  0.37368421 0.0000000
# 4:     2        8.106  0.01151461 0.0000000
# 5:     2        8.106  0.00000000 1.0000000
# 6:     2        8.106  1.00000000 0.2055018

Note that your normalization will have problems if there is only 1 row for a stoer.