This is one of the most frequent questions that comes up on the DTRPG discord server. I can understand the desire, and I was there once as well. You think good reviews will convince potential customers who are sitting on the fence to give your game/adventure/whatever a go.
While that is true, I do not think that chasing reviews is a good investment in your time and effort.
Things I have tried in the past. Every one of these worked, but upon reflection, I decided I would rather not be doing it.
The first was to put a paragraph at the back of each PDF, thanking the reader for buying the book and if they enjoyed it asking them to leave a review. I would then link to the products page on DTRPG. Setting it up was a bit fiddly. You create the product page, save it, get the page URL, edit the PDF, save that, and then upload it.
This is a sure-fire way of reaching the people who have a copy of your book and have reached the end of it.
The second technique was to use the DTRPG email your customers’ tool. This allows you to target people who bought specific products. I would wait two weeks after a product launch, and then email those people who had bought it and asked them if they had enjoyed it. If they had, then would they be kind enough to leave a review.
The benefit of this one is that you can experiment with the messaging and refine it.
Why I don’t do this is that asking for reviews looks really spammy and rather needy. Just because someone read your book does not mean that they enjoyed it. Sometimes there is a mismatch between expectation and reality and the customer could be really disappointed. There is nothing wrong with your book but it was not what they needed, and they may feel that they have wasted their money.
Unhappy people are more inclined to want to vent their disappointment than happy people are to want to praise.
If you spent your hard-earned money on something and it does what you expected, you have already rewarded the creator. If they then start asking you to do more for them, they are asking you to work for them for free. What is in it for the customer?
If you spent your hard-earned cash on something and it disappointed you, and the creator has your money, you are not getting it back. If they then want you to do their marketing for you, you have an opportunity to tell them exactly what you think.
Even if you got 9 out of 10 good reviews, it is more likely that the tenth negative review is the one that people will remember.
I think that as a ballpark figure I get one review for every 200 sales. That may not sound a lot, but they are all organic, and unsolicited.
You could game the system by sending complimentary copies to ten friends and having them all write glowing reviews. The problem with that is that DTRPG customers are specialists in their field. If you have sold less than 51 copies and yet have 10 5 star glowing reviews, that looks incredibly dubious, doubly so if they are all within three days of each other.
If a real customer buys from you on the basis of those reviews and is disappointed, they are quite likely to add a real review that tries to address the balance, in other words, you have provoked a negative review by your own actions.
I think that trying to hunt for reviews is a poor use of your time, and as a trend, reviews are losing their legitimacy. Fake reviewing is a real thing and customers are aware of it.