how do i apply a function within a group that depends on which row within the group holds a value?

You can use which or better which.max which is guaranteed to return only 1 value.

library(dplyr)

df %>%
  group_by(ID) %>%
  mutate(which = which.max(value) * +(row_number() == 1))

#     ID level value which
#  <int> <int> <int> <int>
#1     1     1     0     3
#2     1     2     0     0
#3     1     3     1     0
#4     2     1     0     2
#5     2     2     1     0
#6     2     3     0     0

+(row_number() == 1) is to ensure that the value of which is assigned to only 1st row in the group and rest all the rows are 0.