Generally, the first Sola listed is "Sola Scriptura". Does it matter whether or not we teach our children this vital doctrine?

Yes, this doctrine if crucial to a real relationship with Christ, the Savior.

While we were at my sister-in-law's house last weekend, she told us about her pastor's daughter. She went to Wyoming for the summer; her college set up a job, the dorm, everything...all four (4) of her roommates are Mormon. And yes, they "witness".

Would my kids know how to defend Christianity of the Bible against Mormon, Catholicism, or Jehovah's Witness? Can they use the Bible to defend their faith without the catechism, etc.?

Can they start by explaining why the Bible is the only infallible rule of their faith and conduct?

What is "Sola Scriptura"?

The Protestant position is that all things necessary for salvation and concerning faith and life are taught in the Bible clearly enough for the ordinary believer to find it there and understand.

There are those who fight and strongly protest this doctrine, that Biblical doctrine is all we need in order to find Christ.

The Bible is the Word of God. Jesus is the Word; John 1:14 "And the Word became flesh and dwelt among us, and we have seen his glory, glory as of the only Son from the Father, full of grace and truth."

What does the Bible say about its authority?

John 5:39
You search the Scriptures because you think that in them you have eternal life; and it is they that bear witness about me,

Acts 17:2
And Paul went in, as was his custom, and on three Sabbath days he reasoned with them from the Scriptures,

Acts 17:11
Now these Jews were more noble than those in Thessalonica; they received the word with all eagerness, examining the Scriptures daily to see if these things were so.

Acts 18:28
for he powerfully refuted the Jews in public, showing by the Scriptures that the Christ was Jesus.

Romans 15:4
For whatever was written in former days was written for our instruction, that through endurance and through the encouragement of the Scriptures we might have hope.

1 Timothy 4:13
Until I come, devote yourself to the public reading of Scripture, to exhortation, to teaching.

2 Timothy 3:16
All Scripture is breathed out by God and profitable for teaching, for reproof, for correction, and for training in righteousness,

And what about "tradition"? The word "tradition" is used 16 times in the ESV, most of which deal with the way that tradition is abused:
Matthew 15:6
he need not honor his father.' So for the sake of your tradition you have made void the word of God.

I believe that it is clear that the Bible says that it is in Scripture that we find the Words of God, the Truths of Christ and the Way to the life everlasting.