have somebody do something means that you arrange for somebody to do something. Here are some examples.
I will have the gardener trim the hedge next week.
You need to have somebody fix that step- it's dangerous!
I could have my husband drop by with your prescription if you like.
It does not imply let or make. get somebody to do something generally means persuade somebody to do something:
How can I get my kids to eat vegetables?
get somebody in to... has a roughly equivalent meaning, but could only be used when employing somebody, most likely on a short term basis, for example a builder:
You need to get somebody in to fix that step- it's dangerous!
Regarding your two sentences, the first one suggests that you normally or habitually arrange for him to deliver the letters, or you have arranged that he will normally or habitually deliver the letters. For example you might be talking about a member of staff who is responsible for delivering all of the mail on a daily basis.
The second one means that you recently arranged for him to deliver a particular collection of letters, and he is doing it right now.