- Can I work without Having an SSN?
You can start work without having gotten an SSN as long as you have applied for an SSN. See this brochure:
Do I need to have my number before I start working?
We don’t require you to have a Social Security number before you start
However, in your case, it seems that you are having trouble applying for an SSN, which could be a problem.
- I went to SSN office and they told me that they need a valid i94 to apply SSN.
How to proceed with the SSN for the new card?
This is a tricky situation. In principle, it should be possible to apply for an SSN in your case, since you are authorized to work for 240 days after your I-94 expires during a pending H1b extension, and someone who is authorized to work should be abler to apply for an SSN.
However, I cannot find anything in the SSA manual page for statuses that allow employment, or anywhere else in the manual section on SSNs that mentions the work authorization for 240 days during a pending extension. (And I suspect it is extremely rare for someone to apply for an SSN during the extension period, as in almost all cases, people would have already gotten an SSN during their previous H1b I-94 period.)
You can cite the section of regulations that says you can work. 8 CFR 274a.12(b)(20) authorizes continued work for the same employer for 240 days with a pending application, for several categories of work-authorized aliens including 8 CFR 274a.12(b)(9) that includes H-1. Or, you can show them the section of the USCIS Handbook for Employers on H1b which describes how you can continue to work for the same employer for 240 days during a pending extension. But I'm not sure these will help you get an SSN from the SSA office if their own manual does not mention how to input it into the system in this situation.