Every major CA runs a trusted timestamping service. Mozilla doesn't need to maintain their own timestamping infrastructure, they could delegate to one of the CAs, probably based on some sort of formal agreement with them.
Though the way things stand, all CAs have no problem timestamping sigs made with certs that are from other CAs, so perhaps even no explicit agreement is required.
The checker just has to have roots tracing to timestamping CAs.
I really don't understand why Mozilla designed their system like they did. Code signing is well known and probably even done for the Windows installer of Firefox, why did they just not duplicate the model? Checking expiration vs current time makes absolutely no sense for code signing, esp. at runtime (it could kind-of sort-of make a little bit of sense at install time, but I'm not really convinced, and maybe it is even not really possible to distinguish between the two with an effective boundary)
Though the way things stand, all CAs have no problem timestamping sigs made with certs that are from other CAs, so perhaps even no explicit agreement is required.