OP, please vaccinate your child, vaccines are the reason we have removed much of the risk of previously common disease that killed so many children.
There was speculation, years ago, driven primarily by one doctor, who was subsequently struck off that the combined MMR vaccine could cause issues (such as autism) - however that was conclusively debunked a long time ago.
Measles, Mumps and rubella are nasty, potentially fatal illnesses that vaccination prevents, it's that simple.
The Covid vaccine brought back many of the "anti vax" theories - sceptics were concerned by its rapid development and wide roll out - some forms of that vaccine (particularly the Astra Zeneca) did have a higher risk profile than most vaccines commonly administered, but they have now been withdrawn, and still did much good in an emergency situation.
The Covid vaccine was also the first of a revolutionary development MRNA vaccines, unfortunately, those who didn't understand science properly were concerned the vaccines could alter the DNA in us - that was never the case, but a lot of chattering by ill informed people on You Tube gave rise to vaccine hesitancy.
Your partner has obviously read a lot of this crap - and being concerned for your baby - has taken it as gospel, however he is wrong.
Get your child vaccinated - and good luck with motherhood