I can see one case in which you still might analyze the data.
This only works under three conditions:
your paradigm has no variable durations of either stimulus presentations or breaks or others (or you can determine all the times retrospectively) AND
all your fmri scans have timestamps in their header information (which I think is always the case, at least in the raw
file) AND
the pulse sequences and the presentation were always
started at the same time (or the logfile has at least a starting time and the difference between both starts was consistent).
I guess you can answer immediately whether the first condition is met. For the second one, any appropriate software for opening the raw files should be able to show you the timestamps if they exist. For dicom files there are timestamps for study time, series time and acquisition time where the acquisition time is different for every single file (according to this help site from the AFNI program). The third one you might also immediately know but if not you could look into the timestamps of the subjects where you do have the triggers in the logfile and compare it with the starting time of the logfile (if that exists). If you calculate that difference for all subjects and it's always the same you are lucky and can calculate the trigger times for the subjects where they are missing.