Brandoula, welcome! This is a tough issue! I'm not sure there is any easy way to convince family members regarding homeschooling if they are not willing to learn. It is really hard to break through the paradigm that public school is the only way to learn.
You may try opening a dialogue with your SIL to figure out what it is exactly that she opposes about homeschooling. You could always offer to lend her books if she is open to learning more.
Most of all I would try to remain confident in your decision and trust that she will come around. Have you found some other homeschoolers to connect with? It may help if you are feeling a bit alienated from the family. I don't know if this helps but I am glad that you have come here for support!
BTW, I grew up in Oak Harbor! How are things in beautiful LaConner?