The union has to declare a strike, but that doesn't mean in contentious negotiations that workers don't take it upon themselves to walk off early. And just because you vote down a contract offer doesn't mean you must go on strike either. We had a negotiation 5 or so years ago that when we got the contract info in the mail, the letter from the President of the union said "Voting no means we will authorize a strike without further negotiation with the company". So we voted it down, but only by something like a 55-45 margin. We didn't strike, the union went back and negotiated some more, and we accepted that next deal.
A strike vote is quite often done to give the union the mandate to call a strike if negotiations are going nowhere. But usually, even if that vote is done, a "final offer" is made, and is voted on before a strike happens. It's very rare that a strike happens without a "final offer". But sometimes, as in the case of my employer when we first unionized 25 years ago, negotiations were going nowhere, so we had workers at a large facility walk off the job for a couple hours to get the company's attention, and they fairly quickly started negotiating seriously.