A data.table approach
library(data.table)
DT <- data.table(myvec)
DT[, .(number_of_distinct_orders = length(unique(order_no))), by = name]
data.table v >= 1.9.5 has a built in uniqueN function now
DT[, .(number_of_distinct_orders = uniqueN(order_no)), by = name]