How do I legally use brand logos on my app?

Usually, the agreement by which you are using the API will include trademark-related intellectual property provisions which outline how you many use the provider's marks (logos, placements, etc) in your service, if at all. If there are no such considerations memorialized in the agreement(s) by which you are using these APIs, and you want to use those brands' marks, it would be advisable to raise the issue with the brands during your API integration activities and obtain the proper license prior launch.