I am married and filing our tax separately. My husband doesn't have ITIN number. I am H-1B holder, husband is H-4 visa holder. Turbo Tax is asking for my husbands ssn.

Yes, since you are filing separately it is okay to file without his ITIN.  You would need to override the error and print and mail the return. 

If you expect to receive the ITIN soon, you can file an extension of time to file the return to give you until October 15, 2020.  This may be simpler than filing an amended return later.  You can find the form to file an extension inside of TurboTax in Other Tax Situations or you can file here: TurboTax Easy Extension.

If you have not yet applied for the ITIN, you can also mail your ITIN application and documents or certified copies of those documents along with the tax return to the ITIN processing department.  The ITIN department will then add the ITIN to the tax form before forwarding it on for processing.