Which of the following sentences is correct grammatically?

Neither one is. The order of modifiers is ungrammatical in "a chess beginner's game." It would be "a beginner's chess game." Also, there would be a comma after "game." Finally, while this isn't really an issue of grammar, one would generally say "is an abundance." "Are abundances" isn't ungrammatical, but it's not how people say it, whereas "are abundance" is ungrammatical.

Here's how the sentence would appear were it grammatical:

In a beginner’s chess game, there is an abundance of blunders.

Now, regarding chess beginners, I suppose if there were a league or something and you were referring to a special chess game for beginners, you might say "a chess-beginners' game." That would refer to a special tourney for plural beginners playing chess, but if you're merely referring to the chess game played by someone who is new to the game of chess, then you'd say "a beginner's chess game."

In a chess beginners' (apostrophe after the 's') game there is an abundance of blunders.

The first is, but you might consider replacing "there are abundances of blunders" with mistakes or errors are common.

Both wrong. "In a beginner's chess game there will be an abundance of blunders."

I would say that "The game of a chess beginner has an abundance of blunders". In a chess beginner's game is ambiguous as written. Probably should be "a chess-beginner's game". With this type of player, their games could have abundances of blunders, I suppose; each game could have an abundance, so there would be abundances in many games.

Neither sentence feels like normal English to me. I am a 71-year-old native user of British English. I might say: "Chess beginners make lots of mistakes", or maybe: "People who are starting to learn chess make many mistakes in their early games".

If those sentences are in your textbook, it is time for you to seek a good text book, one written by a native speaker of English.

In a chess beginner’s game there are an abundance of blunders. or

In a beginner's chess game there is an abundance of blunders

In a novice beginner's chess game, there are an abundance of blunders.

'Abundance' is singular. There is an abundance of blunders.

And I agree with Billy, it would be a beginner's chess game.