I like the answer by @DimitrisRizopoulos. Here's another approach.
Fit a random-intercept only multilevel model to the shooting percentage data. The data should be in long form. Each row represents the outcome of a single trial, success/failure. So some individuals repeat on several rows, others appear for one or two rows, player 4 will appear on four rows, three of them with outcome 1 and a fourth with outcome 0. It can be a multilevel logistic or linear regression, don't think it matters very much, see comments at the bottom.
What you want to estimate here are the random intercepts for each individual. Individuals with a lot of measurements will have pretty much the same scores on the probability scale. Those with fewer measurements, we choose to believe them less, so their scores will be shrunk towards the grand mean of everyone. Then take these random intercepts as your x going forward to use in your next regression for your actual outcome.
This approach is not without problems. For one, the random intercepts you will pull out of the model are estimates only with their own uncertainties. But it is one way of expressing skepticism about the scores of individuals with very few measurements.
Additional issue, if using multilevel logistic, you can choose between using the random intercepts on the logit scale versus the random intercepts on the probability scale. If using the logit scale, you can use the random intercepts directly. If using the probability scale, you will first need shift the random intercepts by the intercept of your multilevel regression before the inverse-logit transformation to probabilities. Both approaches make conceptually different claims about the relationship between your ultimate outcome and this shooting variable. You can test which approach better satisfies linearity assumptions for your regression.