Do remittance providers require exclusive agreements with their international partners or agents?

No. Unless of course you're Western Union or Moneygram (in which case, you will have to choose one). But as a general rule, no exclusivity required. Tie-ups (the technical term/jargon in the remittance industry) can be many for one agent. You will however, dilute your earnings, etc. if you have more than one tie-up for the very same geographic region. Your tie-up may not channel business your way because of this.