Here is a solution with sqldf
library("sqldf")
myvec <- read.table(header=TRUE, text=
" name order_no
1 Amy 12
2 Jack 14
3 Jack 16
4 Dave 11
5 Amy 12
6 Jack 16
7 Tom 19
8 Larry 22
9 Tom 19
10 Dave 11
11 Jack 17
12 Tom 20
13 Amy 23
14 Jack 16")
sqldf("SELECT name,COUNT(distinct(order_no)) as number_of_distinct_orders FROM myvec GROUP BY name")
# > sqldf("SELECT name,COUNT(distinct(order_no)) as number_of_distinct_orders FROM myvec GROUP BY name")
# name number_of_distinct_orders
# 1 Amy 2
# 2 Dave 1
# 3 Jack 3
# 4 Larry 1
# 5 Tom 2