The short answer is yes.
The Trojan War took place around 1190 BCE. The blurb that the legend is 28 centuries old is more or less correct; Homer wrote about four centuries after the event.
Could Agamemnon have assembled an army of tens of thousands? Yes, he could have. We have many instances of Bronze Age and Iron Age societies able to assemble such numbers.
Hundreds of ships is another matter, but given the shipbuilding technology of the time, plus money, it was possible. The Acheans were a seafaring people. Indeed, this was probably the root cause of the war - conflict of trade between Greece and the Black Sea; control of the Hellespont.
Supporting such an army would have been difficult, but not too hopelessly complex. A stationary army cannot forage for long but Agamemnon's army is on a beach, with plenty of ships. He could build a wharf and ship supplies straight from Greece. There are no tides in the Aegean so it would be straightforward enough.