Prove data points not enough? Estimate required number of data points? I am working on a problem with only ~ 30 data points, I need to use these 30 points to build a machine learning model (ridge regression, random forest ... etc.), then use it to predict on another thousands of data points.

After some experiments, I believe the data set is too small. However, I am wondering is there a standard approach to prove that whether we have enough data points? And how to determine how many data points are needed? Thanks!
It depends on what you mean by "needed". More training data = lower variance with the same model complexity. So when you have a small training dataset, your models will have to be very simple (like linear regression), so to have best generalization. And very simple models tend to badly represent reality.

The answer is: you need so many points, such that your validation error is below desired threshold. 30 is probably much too low for any more complex machine learning model than linear regression with one, maybe two features.