They have an army...
They're just not at war with anyone.
A bunch of helicopters showed up and dudes with guns jumped out, shot anyone in their way, grabbed the president and his wife, and jumped in another helicopter and flew away.
There was opposition, at least one US helicopter was shot down and "a few" Americans were killed.
But it's not like anyone saw this coming, because not kidnapping the ruler of a sovereign country is a pretty big thing among world leaders.
Like, you might as well ask why no one stopped Luigi, the target had no idea this was gonna happen until it was happening.
They didn't shoot the helicopters down instantly, because that would escalate to war.
But every single world leader has heard about this, and when/if they see US troops now, they're a lot more likely to shoot first. Because now it's a very real possibility that they will personally be captured, even if there's no war.
Edit:
And to clarify, it was just a kidnapping and not an invasion.
They left no troops and no attempts to fill a power vacuum or for peacekeeping.
For an invasion, troops stay and there's something to hit back against.
And I think there's a chance they took him to NYC so if anyone tries anything to free him, trump blames Mamdani. It would make way more sense to take him to DC.