Who is responsible for updating a soldiers apft card? The soldier or his Company's S1/orderly room?

There answer is, "Yes." Both. The unit should maintain good records - but often don't. Ultimately, the Soldier is responsible for their records (if they value their career). Keep a copy of everything from your flu shot paper to every training and most recent events (APFT, weapons qual, etc). At the end of the day, if the training room loses something, and you don't have proof that you did it, the onus is on you to prove it.

It'll vary by unit SOP, but in my unit it is the NCOIC of the APFT, partnered with the Training NCO for data entry into DTMS, overseen by the First Sergeant to ensure integrity.
The S-1 administers FLAGs at the direction of the commander once they are counseled post failure.

Works great.

The training room should have access to eMILPO for updating the APFT and weapons qual. They should also have access to DTMS. It will take a long time if you are starting from the bottom and just learning the system but overall a great tool to keep track of stuff like this. As already stated I would request a copy so that you can take it to the S-1 shop and get it updated.