In your article on making berakhot for the COVID vaccine, you mostly support שהחיינו and הטוב והמטיב but also say that הגומל makes sense for people such as healthcare workers who have been more exposed to the “front lines” and therefore feel more a sense of rescue rather than simple relief or joy. If someone has that emotional attitude to the vaccination, should they say הגומל by themselves at the time they get the shot, or in public the way standard cases of הגומל are said?
The first question is whether this person feels that she has been saved from being in a dangerous situation or has avoided danger. If it is the former, then she can say HaGomel. If it is the latter, then she cannot. See my teshuva on this topic.
As to saying it at the moment or waiting until there is a chance to say it in public – I don’t see why this should be different than any other recitation of HaGomel. Although there is a debate in the poskim whether 10 are needed bi’dieved, there is no reason here to get into that questionable situation. I understand the emotional need to say something at the time – the person can always say it without שם ומלכות. But she should wait until there is a gathering of 10 before saying it as an actual brakha.