How many participants are needed for online reaction time studies?

There are four variables of interest in a statistical power analysis: power (often chosen as 0.8 or 0.9 by convention, this is the probability of correctly rejecting the null hypothesis if it is false), tolerated false-positive rate (alpha, often chosen as 0.05 by convention), sample size, and the effect size (which in turn depends on the variability of the observations and the difference between groups).

When you do a power analysis, you can choose any one of those variables to calculate provided you have all the others. In your case, you want to know the sample size you need so you need to choose or estimate the other numbers.

If you know what statistical test you are using, there are numerous online calculators to help you choose a sample size. Personally, I like to use the free software G*Power.

Power and alpha are typically chosen by convention, so what you still need to know to determine your necessary sample size is the effect size you are looking for. If you have an experiment that you expect to have a large impact on reaction times, you don't need as many participants. If you are looking for a subtle effect, you need more. You can choose that threshold based on what is reasonably interesting/clinically relevant (this is ideal but sometimes frustratingly difficult to settle on), or you can base it on pilot data (can be dangerous when pilots are based on small sample sizes).

If you don't have any preliminary data, the best way to estimate the variance in observations would be to look at other similar studies collecting online reaction time data.

Overall, if you want to make sure you have a sufficiently powered study, it's good to use conservative estimates. If you guess that your effect is smaller and the variability is larger than what you hope, then you have a better chance of a sufficiently powered study.