No, a member FI type does not need to wait until the lead FI has obtained a GIIN before it registers or obtains a GIIN. Once the member FI account has been created by the lead FI, the member FI can complete and submit its own registration independently. The member FI registration processing and GIIN assignment are not dependent on the lead FI's registration status or GIIN.
Sources:
Publication 5118 (07/2021)
Notice 2013-43
Instructions for Form 8957 (06/2018)