I tend to think so. For the most part, the act of foot washing, which is based on Jesus' actions in John 13, contains every criteria that Christian faith has traditionally used to define the sacraments.
First of all, it involves action. The act of foot washing originates in the action of Jesus and therefore finds its meaning in action. Sacraments must be performed. When they are not performed, sacraments cease to be (i.e. they do not exist outside of their being performed). Like Baptism and Eucharist, the act of foot washing requires our action.
Second, foot washing contains a visible, outward symbol (water, feet, hands, etc.). In The Institutes Calvin writes, "It seems to me that a simple and proper definition is that it is an outward sign by which the Lord seals on our consciences the promises of his good will towards us..." (IV.xiv.1,3) Hugh of St. Victor defines a sacrament as follows: "A Sacrament is a physical or material element set before the external sense, representing by likeness, signifying by its institution, and containing by sanctification, some invisible and spiritual grace."
Third, the act communicates. Peter Lombard, who solidified the seven sacraments of the Roman Catholic Church, wrote that the sacraments are "for the sake of sanctifying, as well as signifying." The act of foot washing unambiguously communicates grace. It is a 'speech-act' that proclaims the gospel in and through action.
Fourth, the act is linked directly to the Word of God. Foot washing finds its origin in the action of the Word-Incarnate, and in Jesus' command to "wash one another's feet." Regarding baptism Martin Luther said that "it is the Word of God with and through the water" that makes it a sacrament. I see no reason that the same cannot be said of foot washing. The fact that the Word of God commanded this act to be repeated suggests that Jesus' own presence might remain in the act of foot washing.
Fifth, like Baptism and Eucharist, foot washing is a unique and repeatable act. Like other traditional sacraments, foot washing has its origin in a unique, historical event and possesses the potential to be repeated in the ritual of faith.
The only reason that I can imagine to disqualify foot washing from being a sacrament is that the presence of God is located in the human being rather than the element. That is, unlike Baptism and Eucharist where the presence of God is located in the elements, the presence of the divine in the act of foot washing seems to be located in the person washing the feet. For some, this might seem to put human beings in the place of God.
But doesn't Jesus command his disciples to become means of God's grace when he tells them to "do as I have done to you." And what about in John 20:21 when Jesus says, "Just as the Father sent me, I send you"? And doesn't Paul proclaim the idea that Christians are ambassadors on Christ's behalf, carrying out the ministry of grace? (2 Cor. 5:19-20)
That the act of foot washing might convey human beings as the means of God's grace does not disturb me for a second; not when Jesus asks his followers to become agents of grace and "do as I have done to you."
Yes, I think that foot washing could and should be a sacrament in the Christian Church. Like marriage, foot washing is scandalously relational and cannot be watered down into an inward, individualistic ritual (as the Eucharist so often has). In an age where we all struggle to build loving relationships, perhaps the washing of feet is a way to living relationships in the Way of Christ. But if foot washing is to have any meaning at all, it must be performed in the context of the community of faith; that is, in the Church. And there's really no better way to do this than to make it a sacrament rather than a once-a-year ritual.