Net Promoter Score
You can use the Net Promoter Score to get more reviews. First of all ask a similar question like for example: How likely is it that you would recommend my brand/product/service to a friend or colleague?
When the customer rates this question, you could ask a second question:
What’s the most important reason for giving us that score?
If your user gives you a score of 9 or 10, you could send them an automated email asking them if you can use their comments in a review.