Especially with proper priors (which again, everyone running an A/B test should have at this point) you don't have to "plan a sample size". You work with the data you have, and assess the risks in the decision give what you know. There is likewise no risk of "early stopping".
If you're doing Bayesian analysis and only have say 100 visitors, and are using priors, the only situation in which you'll have a strong posterior is when the winning variant is notably superior to the current one. All of the ad hoc rules Frequentists put in place are not necessary when you do proper analysis of your posterior probability that A > B.
Again we see where Bayesian vs Frequentist doesn't really matter for a clinical trials but does for running an A/B test. In a clinical trial you are going to have to plan how many people are involved anyway, so doing a test power calculation makes sense since you'll need some number and that's a good way to get one.
For running an A/B test the bigger problem is usually time not numbers (an exception being email campaigns in which both tend to matter). In the Bayesian setup you if your boss says "I need an answer tomorrow morning" you can take the data you've got, show the probabilities of improvement as well as the distribution of the risks if you're wrong. This allows you to make riskier moves when it makes sense, and be more conservative when it does not. This is something that the out-of-the-box NHST does not allow you.