Fer sures:
Catalytic converter. Single exhaust.
EGR
Duraspark Ignition
Fully enclosed air cleaner housing with cold-air snorkel to the corner next to the battery. No holes drilled into it for "extra air flow" or it will fail most visual inspections.
Vacuum/temp controlled hot air tube from the exhaust manifold to the snorkel.
Hot air tubes from the exhaust manifold to the carburetor choke
Evaporative system, including the plastic charcoal canister on the firewall.
Lots and lots of vacuum tubes going to unknown dark areas. Including:
Front, back and bottom of said air cleaner housing. Distributor advance mechanism, at least two, if not three "timed" temperature controlled vacuum switches in various parts of the intake manifold and the air cleaner housing and such.
Some carbs had a float bowl vented to the charcoal canister. Some didn't. By '77 I'm not sure if they all did, or just CA, or just what.
Maybe for sures, but maybe nots:
Thermactor system. As mentioned, not all EB's had them. Even in CA they varied from year to year starting in '66.
Dual-diaphragm distributor maybe. I've heard that in '77 they went back to a single diaphragm, but don't have a verification on that rumor.
Maybe a carburetor with float bowl vent running to the charcoal canister.
Allowable upgrades in many jurisdictions:
Headers as long as they do not cause the removal or defeat the purpose of existing smog control devices.
Aftermarket air cleaners as long as they're sealed to under-hood areas, include the cold-air intake function, and all vacuum and heat control devices are still functional.
Carburetors? Usually no changes allowed in most areas. Some might allow a 4bbl if the connections and fittings are the same as the smog carb, but most places don't want you to mess with that.
That's about all I can think of. Your basic '77 would have had a crapton of stuff on top of the engine. Mostly connected by vacuum lines, but several connected to electrical circuits which you hopefully still have as part of your wiring harness.
At the very least, do your best to connect vacuum lines to whatever they're supposed to fit, and hope you don't have too much missing. For Early Broncos, around here at least the books are not so detailed in what exactly is supposed to be under the hood as later trucks and most passenger cars. So if it "looks" like it's all there, you might just get lucky and pass the visual tests even if a few small components are still MIA.
The real proof is in the sniffer if they do that. That's where a lot of missing stuff and hotter cams tend to show their colors.
Good luck.
Paul