Okay, here is another more row-wise approach. Let me know if this is more in line with what you hoped, @Enuma.
tibble(E1 = rnorm(10), E2 = rnorm(10), E3 = rnorm(10), E4 = rnorm(10)) %>%
mutate(row_sd = pmap_dbl(., lift_vd(sd))) %>%
mutate_at(vars(E1:E4), list(norm = ~ . / row_sd))
Or if you want to remove the original E1:E4 variables:
tibble(E1 = rnorm(10), E2 = rnorm(10), E3 = rnorm(10), E4 = rnorm(10)) %>%
mutate(row_sd = pmap_dbl(., lift_vd(sd))) %>%
transmute_at(vars(E1:E4), list(norm = ~ . / row_sd))