Is it possible to request Familysearch to obtain certain records?
I can think of many records that would help me and other genealogists a lot, but they're not to be found on the site.
FamilySearch cannot obtain every record, nor does it try to. Increasingly FS is forming partnerships with other companies rather than trying to get everything on their own website. For example, FS has a partnership with Ancestry.com. FS can only publish records with the permission of record owners, and they may not always be willing to let them. The ongoing pandemic makes it difficult for FS to send volunteers to archives to scan records.
If you name the records you are interested in, a FS staff member will hopefully see this and investigate whether they can be published. Other forum members may know more about why the records are not yet on FS or other websites where they are.4
Well, I was thinking about the census records and church books around Saint Petersburg that are kept here: https://spbarchives.ru/0