Is there is a list of which translations of the Bible are accepted by the Southern Baptist Convention?
The Southern Baptist Convention, while the highest authority within the Southern Baptist church, is not technically an authority amongst Baptists. It functions as a voluntary association with which member churches may choose to associate, but cannot dictate terms to anyone. While it does approve changes to the "Baptist Faith and Message," these are to be taken as guidelines not at odds with the local autonomy of the church.
Within the Baptist Faith and Message (the closest thing Baptists have to a creed), it simply says:
The Holy Bible was written by men divinely inspired and is God's revelation of Himself to man. It is a perfect treasure of divine instruction. It has God for its author, salvation for its end, and truth, without any mixture of error, for its matter. Therefore, all Scripture is totally true and trustworthy. It reveals the principles by which God judges us, and therefore is, and will remain to the end of the world, the true center of Christian union, and the supreme standard by which all human conduct, creeds, and religious opinions should be tried. All Scripture is a testimony to Christ, who is Himself the focus of divine revelation.
The BFM itself is silent on which versions are "approved."
That said, I suspect the question wants to know what translations are preferred. Here, any major translation would suffice, I propose the following list to help you figure out which ones are most likely:
Conservative to Evangelical: English Standard Version (ESV), NIV, HCSB (if you went to Dallas Theological Seminary)
Moderate to Scholarly: NRSV, RSV
Moderate to Gender-inclusively liberal: TNIV
Super-traditional: King James Version. Maybe New King James, if they're feeling special.
Conservative but Scholarly: The Amplified Bible.
When the Preacher wants to make a point: NLT, "The Message"
Perhaps more useful would be the ones Baptists would be very unlikely to use:
New Jerusalem Bible, Duoay-Rheims: perceived as too Catholic
New World Translation: restricted to the Jehovah's Witness
Makes most conservatives barf: The Living Bible, Readers Digest Condensed, Good News Translation.
Televangelist Southern Baptists like Charles Stanley and Ed Young use the NASB translation. They are former presidents of the Southern Baptist Convention. This does not make the NASB the SBC's official bible, but it shows a preference.
What about the NASB?
The SBC does not support the TNIV nor the 2011 revision of the NIV because of the inaccurate translations pertaining to gender. There are resolutions stating such.
They were fine with the 1984 NIV, but it is no longer being printed. They now support the HCSB.
During my time as a Southern Baptist, i gravitated toward the NASB, NKJV, and the ESV. The church to which I now belong uses the NRSV, although I am still partial to the ESV, which is very close to the NRSV.