steam_api.dll
This file is not a system component: supplied with the game by steam. Restore it from the source that installed it rather than downloading a replacement.
If the file is already there and Windows still complains, read the message again: “entry point not found” is a different problem from a missing file, and installing the runtime again will not solve it.
And if the box says RUNDLL rather than naming a program, the fix runs the other way: something left an instruction behind pointing at a file that is meant to be gone, so restoring the file is the wrong move.
Do not download this DLL on its own
Searching for a missing DLL by name leads to sites that hand you that single file. It is one of the oldest and most effective malware delivery methods there is, and a system DLL from an unknown source is indistinguishable from a deliberately modified one. There is never a legitimate reason to obtain a runtime component that way.
The same goes for the tools that offer to do it for you: what a DLL repair tool can and cannot actually fix.
What it is
The Steam API library. It is not a system component: it ships inside each game folder and is installed by Steam along with the game.
Where you see it: A game that fails to start after an interrupted update or a file that got quarantined.
If it is missing from a game you own, use Steam's Verify integrity of game files and it will be restored. It is worth knowing that this error is also extremely common with unauthorised copies, which is why so many sites offer the file directly. Downloading a replacement steam_api.dll from a random site is one of the most reliable ways to infect a machine.
Why there is nothing to download here
There is no verified download we can offer for this one, because installing a file is not the fix. The steps above are what actually resolves it. That distinction matters: the sites ranking for this file name are selling a download that would not have helped even if it were safe.