At my last job, pretty much all our hospitals stopped giving us face sheets, don't even ask for one, you won't get it.
At least hospital I heard (through the grapevine rumor mill) that it was because they had found a facesheet or two (or 10) in the ambulance parking area....i.e. it blows off a gurney or out the open window and crews either didn't notice/didn't care and the hospital said "HIPAA" in response to the sheet being on the ground in the biblically accessible area....but that was the rumor mill reason I heard (Though plausible IMO) for one out of a dozen or so different hospitals that wouldn't give us face sheets so IDK. Kaiser would give them, you just had to ask...but as everything else became electronic that became one of the only paper documents left we'd have to keep track of and log and turn in, so a lot of guys simply stopped asking Kaiser for facesheets....unless we couldn't get all the patient info from the patient/family and they (Kaiser) had it then it was nice.
As far as documentation without a face sheet? Simple, we asked the patient themselves! We'd often just ask to see their ID, and 9 times out of 10 we'd then copy the info off of their drivers license. Otherwise we ask the patient (or family), we'd get name, birthday, and phone number were critical must have fields, address was easy enough to get....health insurance info, most people could tell you if they had it and who it was (or that no they didn't have insurance...) fortunately the number wasn't critical, and if they happened to have the card available, perfect. Otherwise billing got to deal with it. I've had a couple patients where we treated and transported Jon Doe, unknown birthday, estimated age, unknown address, phone, and, insurance info, pt unable to provide info due to the fact I'm BVM'ing him on the way to the trauma center after Pt was found down in the roadway after being hot by a car.....
We'd get the MRN (all patients, whether it was a John Doe or someone we got every box filled)....required to clear the hospital and go back in service, even with calls pending...we were also supposed to get the name of the Attending Physician for the pt, supposedly for MediCare/MediCal. Sometimes that was easy, it was the guy/gal who you have the report to and signed the paperwork, otherwise I asked the nurse, and sometimes Dr "None Assigned Yet at time of transfer of care" worked at quite a few of my hospitals lol.
Long, somewhat rambling story short, your company will have different policies regarding face sheets than mine did, best to ask your Supervisor, but chances are your hardly the only one to not get one, highly unlikely you'll get fired for that (unless you chronically don't have them and have been councelled in the past about it). Just make sure you have sufficient info from the pt themselves on your run form that billing can still do their job and you'll be fine. But yeah, just don't make a habit of losing the facesheet.