Unless you are intentionally making use of a trademark that you do not have permission to use, all they can do initially is just ask you to remove/change whatever it is causing the issue.
Are you asking a question about the use of a company logo within your app? Generally speaking, large distribution websites like amazon and ebay provide a collection of "marketing" material for use within apps. This is because any links into their site would be beneficial for them. On the same page, you can normally find information regarding contracts of usage.
I found this page on Amazon:
https://www.amazon.com/gp/help/customer/display.html?ie=UTF8&nodeId=201713630
So long as you reference this in your google play store application process, you should be fine. This is because amazon is actively giving you permission to use their logo, and so long as you comply within amazon's provided guidelines, and make google aware that these guidelines exist, you should be fine.