It all depends what is required as documentation to prove vaccination for travel. I can print from my doctor's patient portal pretty much anything including a list of immunizations I've received (in recent history). Same for DD's pediatrician. I'm sure my husband's portal has similar. Whether that will be acceptable proof remains to be seen. To date, I can say it has been accepted for school records and camp registrations for DD. I haven't had to prove my own vaccinations for a long time, though I do still have the paper record my mother passed along to me from my childhood vaccinations. But the records and information is likely there, just someone may need to ask for it or locate it.
The bigger question is what can or might be falsified, and that's not unique to Americans. I wasn't familiar with the yellow book until mentioned on this thread, but it certainly isn't fool-proof. I'm sure there are laws or rules about providing false information... whether those get enforced remains to be seen.