Can you get FIFA 2021 for Nintendo switch?

You can...  but you probably shouldn't...

The main reason why I'm recommending NOT buying the Switch version is that EA released it as a "Legacy Edition", which translate to "We want to release an version with an updated roster, but we don't spend any money updating the game core so we can reduce costs & maximize profits"... in short it's nothing more than a lazy cash grab.

FIFA 2021 & 2020 Legacy Editions are based on FIFA 2019.  This means that any bugs you have found in FIFA 2019 will affect 2021 since they're running the exact same game code (only the rosters are different).

It's ultimately your choice if you really want to buy it, but you could just complain to EA to release a legitimate updated version or see if FIFA will discontinue their agreement with EA & go with another publisher (like Take Two & 2K Sports).

Though the Switch versions of FIFA 2021 and 2020 are called "Legacy Versions".

They are pretty much the same game as what was released on the Switch for FIFA 2019 with a roster update.  So don't buy this thinking you'll get the same game that's playable on Xbox.  It won't be the same game.

Yes it came out on Oct 9th for all consoles including switch.