Can Southern Baptists take communion at home alone?

While I am unable to find any evidence that the Southern Baptist Convention definitely supports the practice, there doesn't seem to be any reason for them to do so. For one thing they/we believe in the "priest-hood" of all believers, so there's no requirement that communion be administered or overseen by an ordained minister. In their "Basic Beliefs" web page, they state: "The Lord's Supper is a symbolic act of obedience whereby members … memorialize the death of the Redeemer and anticipate His second coming." Here's a link to that page: http://www.sbc.net/aboutus/basicbeliefs.asp

Each congregation decides how often they will observe communion as a congregation, and while there is never an implied injunction against extra communion services by "sub-groups," the expectation would be that all who partake are verified members of the church by baptism.
On an anecdotal note, my SB minister of many years would often observe a communion for himself alone before he began bible study.

I have been a Southern Baptist for over sixty years, and have never heard of any set procedure; for administration of Communion. In all of the Churches I have attended the juice and wafer are distributed by the ushers and after distribution usually the Preacher offers a prayer; but in some churches everyone says their own silent prayer, then the bread and juice (We do not take wine) are taken.

The one commonality in all of these Churches is that these Scriptures are read by the Preacher:

Luke 22:15 through 20 Then He said to them, "With fervent desire I have desired to eat this Passover with you before I suffer; 16 for I say to you, I will no longer eat of it until it is fulfilled in the kingdom of God." 17 Then He took the cup, and gave thanks, and said, "Take this and divide it among yourselves; 18 for I say to you, I will not drink of the fruit of the vine until the kingdom of God comes." 19 And He took bread, gave thanks and broke it, and gave it to them, saying, "This is My body which is given for you; do this in remembrance of Me." 20 Likewise He also took the cup after supper, saying, "This cup is the new covenant in My blood, which is shed for you.

Without any other guidance from the S.B.C. I see no reason that one would be restricted from following that procedure alone at home, and especially if they are homebound.

There is no "official" position on this question, since the Baptist Faith and Practice is silent on the matter. And in a body as diverse as the Southern Baptist Convention, there is sure to be variability in practice.

That said, it's safe to say that "private communion" by a layperson is clearly outside the Southern Baptist norm for two reasons:

• The Lord's Supper should be observed in the context of the local church
• A minister should serve the Lord's Supper, not a layperson

Both points are rooted in the standard Baptist (and Protestant) viewpoint that the Lord's Supper is an ordinance for the church, not something given to individuals. Ray Van Neste writes:

To describe the Lord's Supper as a church ordinance is to assert that this rite was given to the church to practice and not simply to individual Christians. This is the understanding of the great majority of Christians across the history of the church. ("The Lord's Supper in the Context of the Local Church," 369)

In the Scriptures, Communion is part of the gathered worship. It is not merely a private act. [...] Thus Communion is not fitting in individual situations. (376)

Van Neste indicates that this was also the view of leading Southern Baptists like W. A. Criswell and George W. Truett. He quotes Criswell's Doctrine of the Church (105) as follows: "the pastor should not take the Lord's Supper to individuals and administer it personally."

Similarly, Bobby Jamieson writes:

Only a local church should celebrate the Lord's Supper, and they should celebrate it in a gathering of the whole church. [...] Despite the commendable compassion behind the practice, it shouldn't be "taken" to those who are homebound or in the hospital. (Understanding the Lord's Supper)

These authors assume that even in these "individual" situations, communion is being served by a minister. This is explicitly required in a number of Baptist writings, such as the London Baptist Confession (1689), still highly regarded by many Southern Baptists:

The Lord Jesus hath, in this ordinance, appointed his ministers to pray, and bless the elements of bread and wine, and thereby to set them apart from a common to a holy use, and to take and break the bread; to take the cup, and, they communicating also themselves, to give both to the communicants. (30-3)

Similarly, Millard Erickson writes:

At least some of the duly chosen leaders of the church should assist in the observance of the sacrament; the pastor should take the leading role. (Christian Theology, chapter 53)