The relationship between possibility and proof is complicated, and the subject of much philosophical analysis. A good article in the SEP on the topic is "The Epistemology of Modality". Let's start with the simple:
Carl Sagan who famously discussed SETI and alien intelligence, IIRC, famously coined the phrase "Absence of evidence is not evidence of absence". One cannot search the earth and say, I have not seen alien beings, therefore alien beings do not exist, because to do so deductively leads to a fallacy. Obviously, if we were to search 1,000 planets in the local neighborhood, then we would have a much stronger case, but still an uncertain one because new evidence might defeat our conclusion. In logic, such evidence is called a defeater. Falibilism recognizes there are always things we don't know we don't know.
There is a real world example of this involving white and black swans, and to reason without recognizing that unknown unknowns are problematic for certain conclusions, one can be surprised by black swan events. From WP:
The black swan theory or theory of black swan events is a metaphor that describes an event that comes as a surprise, has a major effect, and is often inappropriately rationalized after the fact with the benefit of hindsight. The term is based on an ancient saying that presumed black swans did not exist – a saying that became reinterpreted to teach a different lesson after they were discovered in Australia.
We cannot prove the negative, but is possibility predicated on lack of disproof? You ask:
Is it a fallacy to say that something is possible if not disproven?
That depends on your views of conceivability. From the SEP article above:
Both Stephen Yablo (1993) and David Chalmers (1996, 2002, 2004, 2010: Ch. 6) have developed independent accounts of conceivability as a source of modal knowledge (for yet another account based on response-dependence, see Menzies 1998). Yablo holds that, when adequately understood, conceivability provides defeasible evidence for possibility.
More simply put, it's rational to presume something is possible if you can merely imagine it.
So, no, it's not fallacious necessarily to believe something is possible if it is not disproven, but it is important to remember that lack of proof isn't proof it isn't possible either. This is of course is a general rule of thumb, and depends on other metaphysical presuppositions.