WebPageTest can’t do this without modifing the source code. However, the most data for each navigation step is already collected. In the _IEWPG.txt file the measurement results for each navigation step are already stored (expect the steps behind logData 0). The server just reads the first step (=line) of this file in page_data.inc (there is a explicit break in the source code).
For the request details its a little more complicated. The _IEWTR.txt file just contains the request details for the last measured navigation step. To get the whole request data you need to change the file creation in the agent code (the files are created in the file results.cc). With the screenshots and video it’s the same issue. Old screenshots are overriden by new ones.
If you just need the results you can use the “raw page data” link on the result page. This generates a csv file, which contains the measurement results for each navigation step (one line = one step).