Strictly speaking you don't need to wait for some arbitrary significance threshold. I don't know why so many people treat website A/B tests as similar to carefully, traditional nhst controlled experiments. Website A/B testing is much better thought of as an optimization problem rather than a true hypothesis test.

What's really important if you want to improve a website via A/B testing is a constant stream of new hypotheses (i.e. new variants). You can call tests "early" so long as you have new tests lined up it boils don't to a classic exploitation/exploration problem. In fact, in early development rapid iteration often yields superior results to waiting for significance.

As a website matures and reaches closer to some theoretical optimal conversion point, then it starts becoming increasing important to wait until you are very certain of an improvement. But if you're just starting A/B testing, more iteration will yield greater success than more certainty.