The 1990 decision of the U.S. Supreme Court in Davis v. United States (110 S.Ct. 2014 [U.S. 1990]) is relevant to this issue, and it resulted directly in a change in Church policy.
Although it was apparently not challenged prior to this, in an income-tax refund suit brought by parents of two former missionaries, the Supreme Court held that funds sent directly to LDS missionaries for their support were not deductible as a charitable contribution and could not be listed as a charitable deduction on their tax return. To qualify as a charitable deduction, IRS rules now stipulate that the funds be given to the Church itself or else donated through a trust or other legally enforceable arrangement, for the benefit of the Church.
Prior to this ruling, general practice was for those supporting individual missionaries to send funds to them directly. Following this court ruling, the policy of the Church was modified. When I served as a Church missionary in 1973, my parents mailed me a monthly check for my living expenses. I never got the impression that qualifying for tax exemptions was ever an issue for my parents.
Since 1990, the Church policy regarding missionary funds is that donations are tendered as a voluntary general gift given to benefit the Church. Missionary Fund donations are made through the same channel as any other donation to the Church, using the provided donation form in the member's individual ward.
Missionary funding is handled through two different channels, but it has the same effect. At the ward level, there is a financial category designated as "Ward Missionary Fund", which the bishop of the ward can make disbursements from at his discretion, but the money is still held in trust at Salt Lake. At the general Church level, the General Missionary Fund serves the same purpose, but the trust is administered by the Missionary Department for disbursement. Either way, the missionary receives funds disbursed from the Church, from Salt Lake City, not from individual donors.
President Thomas S. Monson counseled in the October 2013 General Conference,
To help maintain our ever-increasing missionary force, I have asked our members in the past to contribute, as they are able, to their ward missionary fund or to the General Missionary Fund of the Church. The response to that request has been gratifying and has helped support thousands of missionaries whose circumstances do not allow them to support themselves. I thank you for your generous contributions. The need for help is ongoing, that we might continue to assist those whose desire to serve is great but who do not, by themselves, have the means to do so. (Thomas S. Monson)