I had a similar situation not long ago and the way I've fixed was using memcache and threads.
I've also thought about using Sidekiq, but Sidekiq is ideal if you don't expect to use the data right away, so memcache and threads worked pretty well and gave us a good amount of control.
Instead of calling the API directly I would assign the API request to a thread and this thread once done would write to memcache, in my case this can happen incrementally, with the same API being able to return more data from the same endpoint until is complete.
From the UI I would have a basic ajax pooling mechanism that would hit a controller and check memcache for data and for the status to see if it was complete or not, this would sign the UI that it need to keep pooling for more data.