Yes, since Anzac has a Home Country, it can upgrade its minor factory and move milita and build anything on the Commonwealth’s tech tree, provided that it also has the required infrastructure (such as a Major Shipyard for some types of warships).
If Anzac upgrades to a Major Factory, this also gives an additional tech roll to the Commonwealth.
Remember that “British Commonwealth includes Great Britain, FEC and ANZAC. These share the same turn, war status, technology marker, per-turn limits on technology development, per-turn lend-lease limits, as well as strategic rail and strategic naval movement limits.” (G-B, FEC and Anzac National Reference Sheets).
Only FEC has the restriction that it cannot build units made available to Great Britain after acquiring technology (i.e. advanced artillery, adv. mech., heavy armor, jet fighters…). This does not apply to tech that does not require specific builds (i.e. long range aircraft, radar, adv. sub. warfare, advanced factories…) which benefit all members of the Commonwealth.