Is this a correct sentence?

It is not a sentence, but it's a good start.

You would want to add a 'She was' or 'She is' or similar to the beginning to make it a full sentence. Otherwise, it's a fine.

Well, you need a noun and a verb to form a sentence, so that isn’t one. There’s no verb. “To command” is part of an adjective phrase in this context. But as a form of wording, it’s absolutely fine. It’s probably what I would say. (A full sentence would be, for example, “*She is* the first woman ever to command a submarine,” or, “The first woman ever to command a submarine *was interviewed on television*.”)

The wording is correct, but it isn't a full sentence

It's not a full sentence. That whole thing is a subject but there's no verb. "First to ever command a submarine" is describing "woman" but in order for it to be a full sentence you need a verb. So for example"The first woman to ever command a submarine **was** Jane Smith."