My mother bought a refrigerator and had it delivered.

Literal

Mother [topic-は] refrigerator [object-を] bought-[sequential-て] deliver-[causative-させた]

買って届けさせた chains buying with causing delivery: 買う (buy) + て + 届けさせる (causative of 届ける = to deliver). The causative means she arranged for someone else (the store) to deliver it.