Of course you can vote without being affiliated with a party.
The party affiliation registration only serves to make people eligible for primary voting. In most states, if you want to vote in the primaries, that are the elections that select a party's candidate, only the people affiliated with that party are allowed to vote.
But in the general election, the one that actually elects one of the candidates of all parties, every US citizen is allowed to vote if he satisfyies minimum age and criminal history requirements.
You do not need to state a party affiliation when you register for voting, you can always register as "unaffiliated."