Do finite state machines need transitions for all symbols?
Your question isn't entirely clear but it's easy enough to just list all the possible answers that could arise, depending on exactly what you mean.

If you're asking about nondeterministic finite automata (NFAs) then you don't need to account for every symbol. The rule is that, if you're in a state and you encounter a symbol that has no transition, then that branch of the computation rejects immediately.

If you're asking about deterministic finite automata (DFAs), then it depends on exactly what definition you're using. The classical definition (e.g., in Sipser) is that every state must have exactly one transition for each possible symbol. In that interpretation, you need a trap state, since you must specify what happens for every possible symbol you could read. Other authors, though, say that each state of a DFA has at most one transition for each symbol. In that case, you don't need a trap state: as with an NFA, if you read a symbol that has no transition, you reject.