There isn't an API to get the waterfall itself. Extensions
do have
chrome.debugger, however. This gives you access to the Chrome DevTools Protocol. This is the same protocol that devtools uses to generate the waterfall. I am not familiar with the intricacies of the network tab,
but a while back the devtools team added a Protocol Monitor inside of devtools that can help you figure out what calls do what. It is still experimental, so you will need to enable it manually. You can do it by opening devtools, hitting the ? key on your keyboard to bring up settings, selecting Experiments on the left hand menu, and then searching for Protocol.
Once you enable it, you can click on the devtools menu ( the ⁝ button) > More Tools > Protocol Monitor to add it as a new tab on your drawer. From there, you can see all devtools protocol calls.