How do I proceed with doing my taxes when soon to be ex won't give me her social security number

Ask your divorce attorney to get it for you ... if you are both filing separately then she needs your SS# and you need hers ... do a swap. 

for 2020 it looks like you were married at the end of the year. that means these filing types are available

1) married filing joint

2) married filing separate - if you can get her SSN then you will need to paper file and attach a statement giving 3) her name, address and that you made an effort to obtain her SSN but she refused.  if you filed jointly in 2019 that return should have her ssn

3) if you lived apart for the last 6 months of 2020and paid more than 1/2 the cost of keeping up a home for yourself and your child (child, stepchild or foster child who can be claimed as a dependent) then you can file as head of household and don't need her SSN thus you can e-file with this status.

note that she might file first claiming the child. if she does, you'll have to paper file because the IRS will reject any return when a SSN hsa already been used onan earlier filed return.