While I am not Rabbi David Lau, and I can't speak exactly to the sources that he used to come to this conclusion, we do have some precedence for the halacha (Jewish law) being established according to what the current medical recommendations are.
The G'mara (Kritot 13a) teaches us that if a pregnant women has a craving for tamei food - food that's not kosher, such as the forbidden cheilev, the fat we don't eat from an animal - she is given it, first in small amounts, and then allowed to eat it normally if that's not enough.
The Mishna in Sanhendrin 4:5 teaches us that if someone causes a single loss of life, it's as if they've killed an entire world.
ללמד שכל המאבד נפש אחת, מעלים עליו כאילו איבד עולם מלא; וכל המקיים נפש אחת, מעלים עליו כאילו קיים עולם מלא.
This teaches that anyone who causes the loss of a single life, he is punished as if he caused the loss of an entire world; and anyone who saved a single life, he is rewarded as if he saved the entire world.
(my own translation)
I'd like to take this discussion about whether or not smoking is permitted or not as a source, though, and quote this section (page written by Rav Mordechai Halperin):
מהן ההשלכות ההלכתיות של חידושי המחקר הרפואי?
כבר החפץ חיים בהתייחסו לעישון כתב:
"... ואגב דאתי לידי אודות עישון הסיגארין, נדבר קצת אודות זה. הנה כמה רופאים גזרו אומר שכל מי שהוא אדם חלוש אסור להרגיל את עצמו בזה שמהליש כוחותיו ולפעמים נוגע גם לנפשו. והנה דברתי כמה פעמים עם חלושי כוח אודות זה, וענו לי שגם הם יודעים ומכירים בעצמם שהעישון קשה להם. אך מפני שהורגלו בזה מכבר קשה להם לפרוש מזה. ואמרתי להם: מי התיר לכם להרגיל עצמכם על כך? ! אמת, חז"ל אמרו (בב"ק צ"ב.) החובל בעצמו אע"פ שאינו רשאי פטור (כי למי ישלם אם לא לעצמו), אבל על כל פנים הלא אמרו שאינו רשאי לחבול בעצמו. ראשית משום ונשמרתם מאד לנפשותיכם. [...]
ניתן ללמוד מדברי החפץ-חיים שלושה דברים ברורים:
א. דעתם הברורה של הרופאים קובעת דין הלכתי.
What are the halachic ramifications of new scientific studies?
The Chofetz Chaim in his discussion about smoking writes:
And, by the way, my opinion on the smoking of cigars, I'll talk about that a bit. Several doctors have said that a person who is weak is forbidden to accustom himself to something that takes his strength and sometimes affects his spirit. And I have spoken on several occasions to people who fall under this description of "weak", and they have told me that they themselves know that smoking is bad for them. But since they are addicted, it is difficult for them to stop. And I said to them: Who allowed you to become accustomed to this?! Truthfully, Chazal said (Baba Kama 92) whoever harms himself is exempt from punishment even though he is not allowed to do so (because who would he pay exept for himself), but even so it is expressly forbidden from harming yourself, the first reason being because of "and protect your lives greatly" (venishmartem me'od lenafshechem). [...]
We can learn three clear things from the Chofetz Chaim's words:
1.) The clear opinion of the doctors establishes halachic law.
(my own translation)
While the Chofetz Chaim was writing about smoking, this can be clearly applied to our situation as well. The prevailing medical advice is that wearing masks and getting vaccinated greatly help reduce the chances of infecting someone, potentially fatally; therefore, the halacha dictates that we abide by those instructions.
And, of course, overarching the entire discussion, is the idea of "Pikuach Nefesh", when a life is in danger. When a life is in danger, we do everything within our power to save that life. A pandemic with a deadly disease being spread clearly endangers lives. Millions of people have died due to it. Even if there was some sort of problem with taking a vaccine - if, for instance, it had non-kosher materials, just for the sake of argument - it would be irrelevant. Pikuach Nefesh overrides all other mitzvot (commandments) and aveirot (prohibitons) (with the exception of idol worship, murder, and adultery). This is something we can do to help save lives; that's what is required by halacha, without a question.