(though maybe in some cases - I'd have to check).
It existed, but was more or less informal : Diviciacos was the abassador of Aedui in Rome, but it was more for requesting Roman help and being in Rome as both proof of "alliance" and probably hostage.
As for a modern-looking embassy system, you'd need a complete re-conception of what was the ancient or medieval state, without basing itself on the mental and social reality.
Critically when sending envoys, representatives, in a court strongly implied a submission or at the very least an acknowledgement of inferiority : remember that embassy came from celtic
ambactos, a client.
The lack of quick long-range communication is of course another obstacle. An ambassador you can contact in mere hours, if not less, can certainly represent his country/empire interests and intents.
One that can only have news from his own months away, have a really distinct role : hosting and guiding fellow countrymen, and being send back with directives.