Is it theoretically possible to have a nondeterministic finite state machine without any initial state or does it need at least one initial state?
In my favourite definition a finite state automaton is specified as 5-tuple $(Q,\Sigma,\delta,q_0,F)$, where $Q$ is the finite set of states, $\Sigma$ is a (finite, nonempty) alphabet, $q_0\in Q$ the initial state, $F\subseteq Q$ the set of final states, and $\delta\subseteq Q\times\Sigma\times Q$ the set of transitions. In that definition we would have exactly one initial state.
It is however very well possible that, for reasons of symmetry, one specifies a set $I\subseteq Q$ of initial states. There you can have one, several, of no initial state.
Short answer: consult the book, lecture notes, or scientific paper you are using. They set the rules for the game. Not us, not wikipedia.