Questions about an vs année are recurrent in FSE. See for example Can “an” and “année” be used interchangeably, is there a difference in meaning?
In your specific case, the most common usage is the second one (une fois par an) but the first one shouldn't be ruled out.
If the year it is about is not a calendar year but some period of one year, like une année scolaire or une année fiscale, the first one would be equally possible.
If an adjective like scolaire, fiscale, civile, calendaire, entière, complète... is used, then année is mandatory; e.g.:
"C'est un événement à ne pas manquer, car il n'arrive qu'une fois par année scolaire.
an
that is mostly used (if it can help)