Thank you for the additional information. Let’s say that the amount of time scheduled is 1½ hours for this initial meeting and there are 10 people in attendance. The ice-breaker is geared towards the new group getting to know one another and potentially on-topic.
How about asking them to introduce themselves, describe their own pet, and why they want to be a part of this group? Before they begin, use yourself as an example: “Hi. I am Pied. My pet is Bean, a grey kitten I found as a stray. I want to form this group to help…” By going first, you set the tone and provide an example of what is expected. It also helps to post these three bits on a flip chart so that the members have a visual reference in order to stay on topic. As each individual responds, thank them for their answer before moving on to the next person.
With this simple exercise, it provides introductions to the whole group, a bit of personal information that relates to the topic, and insightful information about why each person is there. If you go this route, estimate on ~two minutes per person attending. In some cases, it will be less; in others, it may take more. No one is going to complain if the meeting adjourns early.
If you think that you can pull this off, ask the volunteers to either bring in a picture of their pet or e-mail one to you before the meeting. They could then be used in an electronic presentation or printed out and held up when the members are introducing themselves. We used to do this when I ran a week-long training class. The photos were then posted on a bulletin board in our training room so that the members could reference back to them as the group got to know one another.