Does the reflexive form of the verb "schicken" have a different meaning? Can't it be hier: Ich habe ein paar Angebote schicken lassen.
First, the correct form would be:
Ich habe mir ein paar Angebote schicken lassen.
If you omit "mir" ("Ich habe ein paar Angebote schicken lassen."), you had the offers being sent somewhere, but not necessarily to yourself.
So in the German sentence, "mir" is important, and omitting it would change the meaning considerably.
The verb schicken (to send) takes three mandatory completions:
Who is initiating the action, i.e. who is the initiator of the sending-process?
The answer to this question is the subject of the sentence, and it has to stand in nominative case. In you sentence this is the word »ich« (I).
What is it that gets sent, i.e. what is the thing that moves from the sender to the receiver?
The answer to this question is a grammatical object, that has to stand in accusative case. So it is an accusative object. In your sentence this is the nominal phrase »ein paar Angebote« (some offers).
- Where to? or Whom?
To which place is thing sent? or To which receiver is it sent?
If it is a place, then you ask "where to?" (in German: wohin?) and the answer is a prepositional object like »an meinen Arbeitsplatz« or »ins Büro«.
But in your case it is not sent to a place, but to a person, and you ask for this receiver with "whom?" (in German: wem?) and the answer to this question is a dative object. In your sentence it is the word »mir« (to me).
If you omit the mandatory third part, you get an incomplete sentence, and you don't know where to, or to whom the offer will be sent.
If you omit mir, you have to translate your sentence to this:
I let someone send some offers.
So you have no idea, where the offers might go to. Adding the receiver helps to clarify this:
I let someone send some offers to me.