Is it possible for God to forgive my family of their sins for my sake?

Each person is responsible for their own decisions in life, as God created all of us with the power of free choice, and we all have to take the consequences of our choices. But when Jesus was on the cross, He asked His Father to forgive the people that had crucified Him. Imagine that! Praying for the ones who hammered in the nails! That is forgiveness and grace.

As far as your situation is concerned, virtually every Christian on the planet has at least one family member (and some of us have many more!) that has not accepted Jesus’ free gift of salvation and eternal life. So, what you have to do, as we all do, is pray every time you close your eyes, for your family. God cannot force them to accept Him, but He can work extra hard to draw them to Him with His love. So, keep on praying, that they may realise what they are (sinners), and that they need a Saviour, Jesus Christ. This is called intercessory prayer, when you pray for others to be saved. The good thing about it is that God loves each one of them, and wants each one of them to be saved. So, when you pray for them, you are 100% in God’s will.

Luke 23     ³³ And when they had come to the place called Calvary, there they crucified Him, and the criminals, one on the right hand and the other on the left.

³⁴ Then Jesus said, “Father, forgive them, for they do not know what they do.” 

This is often a long process, so don’t give up praying. My father took 62 years of praying for him, before he gave his life to God. God bless you as you pray!