Over the years I have experienced this issue dealt with (and sometimes avoided) by churches in very different ways.
Presently, we have a committee with responsibility for "Caring Ministry" in our church. The chairperson does not have authority to write checks, but does have a great deal of discretion with her budget to revise it to meet emergent needs, and to do so on her own authority. The rest of the committee are not so much decision makers as assistants to her in getting the needed help those it has been directed. Her budget is set and reviewed by the Administrative Council to whom she is accountable. If something comes up that she feels needs to be addressed, she will usually call me for feedback, and then go and do what she wants. (Primarily because she is more generous than I might be inclined to be.) Checks are written by the church secretary who is also our bookkeeper, and then must be signed by one of two people who are authorized to sign checks on the church's behalf. From beginning to end, we can usually respond in 24 hours.
At my last church we had an Outreach Committee that established a set of parameters for us who were in the church office. As Circuit Rider described above, this meant that my secretary dealt with more people than I did. Even if I was there, she did most of the work as she is the one who kept the records on who we had helped in the past. Our church limited help to once every 6 months. On occasion, when I had special knowledge, I could and would override that guideline to provide additional help. Our primary response was to issue a letter that people could take to businesses we had accounts with that allowed them to shop for groceries or other necessary items and bill us. The letter would indicate a maximum amount. The people could spend more, but if they did so they were responsible for anything over the figure stated in the letter.
The church before that had nothing in place when I arrived there. We set up a group of three persons within the church that met to discuss needs. I and two lay persons made up the group. We had a set of guidelines we produced to guide our discussion, but we had complete autonomy to respond in pretty much any way we saw fit. There was a special fund that the church established that people gave to and we utilized it to meet needs. If it was running low we made a general announcement and it was quickly resupplied. Once we reached consensus on any response, we simply contacted our treasurer who could cut us a check that day.
The only time we have ever dealt with cash was at a church where I was the associate and each Church Council meeting an offering was collected of those in attendance that was put into the pastor's discretionary fund. But, as I was the associate, I was never part of the disbursing team.
I was never any place that used gift cards as a means to minister to the poor in the community as an ongoing basis. I've seen these bought and given as gifts to people at specific times, but not kept on hand. I think it is actually a good idea with one enormous concern that might override the positive. Once purchased and on hand, they are as good as cash. Keeping even a few of these on hands is more than most churches would be comfortable having in a petty cash fund. There may also be some loss of accountability, though this need not be so. All of this is for a church to decide as part of the church's overall ministry, and then set in motion through whatever committee will oversee it, be it the PPRC, the Outreach/Missions Committee or some other group the church designates for that purpose. Your pastor may or may not be a part of that process. There are good argument both to include an exclude him/her. Those views, personal preferences, and the overall missional goals of the church need to be discussed to reach some conclusion as to what you corporately want to do, will do, and how you will go about doing it.