Can I pick up my mother's birth certificate if she is not able to come to the office? If so, what do I need to bring with me?

DC Vital Records cannot search for a birth certificate by SSN. SSN is not available to DC Vital Records for persons born in the District. See this page for requirements https://dchealth.dc.gov/vital-records

Yes you can... bring ID ...and your Birth Certificate just in case especially if you don't know their Social Security number. Death certificates can be obtained here as well. I believe the fee is $18. Bring that as well. You can order borh online as well.