@chahunt9,
My first thought was that yes you were overstepping your bounds, however, on second thought depending how you do this - it could be a nice and thoughtful gesture but you need to tread lightly or it could have the opposite impact.
I am guessing these are adult children - not minors? If so, I am also guessing you have their home addresses? Why not send them an actual hand written letter?
Another question I have is how close are they to their father? This could also have an impact on if and how you write this letter.
I usually agree with jespah - but I do disagree that these children are down a so called rabbit whole and will not change their mind. We do not know them and you have not written why and how adamant these children are about getting the vaccine. I know some people who were against getting the vaccine and now have gotten for different reasons overall. People who have decided against getting the vaccine have done so for a variety of reasons. I know people who feel this way and nothing will change their mind and others that may be more open depending on the situation - so I do not write anyone off.
That being said - if you decide to approach them or write them be very careful on how you do so. You should be respectful on how they feel while at the same time explain how it will impact their father.
For example I would not focus on their health and how the vaccine can help them - it is likely that is their fear that the vaccine could have side effects which is a legit concern as every vaccine does have side effects so arguing that there are no side effects or limited will be fruitless. Also if they are young they are likely thinking well it is less likely I will get very ill if I do get covid - which stats are in their favor so arguing that will be fruitless if knowing there are cases where younger people do get very ill - stats are in their favor.
So what do you have that you could implore them - the fact that it would likely really harm their dad - there is no argument about that. Also, not seeing him for a year when he really needs family for support and love during a hard time - now this really would only work if they were very close.
I do think you should let the family (your sil and daughter know what you are thinking) - they however, may know kids better so may realize whether this would actually help or not or whether it causes a greater divide.
Me, I think if I were to write such a letter, I would let them know how much their dad loves them, how life threatening his situation is, and how them being around them would help his recovery so much. That you respect their decision on what they feel comfortable with doing - but to please consider this - that if they are unsure of the vaccine, to speak with their health care provider to get as much facts as possible so they can make an informed decision.
I think in these uncertain times that instead of being judgmental (whichever way you believe) on another's decision that you take the approach of trying to understand someone else's fears can help - I do not think most people refusing the vaccine are doing so because they are selfish - it is more this is an unknown and they are fearful for something that is new and that it appears the government is pushing on them. I have family members that are very divided some that simply refuse and believe that this is being pushed on us all and we are just blindly following the government (don't blame them it is not like our government instills a lot of trust) to the opposite side those that tried to jump the line to get the vaccine over others that needed more protection.
As an aside - the Pfizer vaccine is close to completing the FDA approval process and I know for many this may ease their fears.
I think showing you appreciate their viewpoint and it is just as valid as theirs will at least open the door to more communication - be respectful of them and they just listen a bit more or least seek out additional information.