I am new to Russian, so I am not sure if there is already a word, but if "gamer" was translated to Russian, would it be видеоиграник or видеоигранец?
Игрок. Видеоигрок; Игрок в компьютерные игры, игрок в видеоигры. .
"would it be видеоиграник or видеоигранец?" - no, but it sounds funny :> "видеоиграник" - as if video games had turned him into something like cubic-polyhedron or some other strange geometric shape ... :> Заигрался... до многогранника :> ... "видеоигранец" sounds really satirically,scornfully :> Видеоигранец-задранец... :>> You really could use these two neo-words for mocking :>
There are indeed multitudinous agent noun suffixes (суффиксы имён деятеля) in Russian, like -ец, -ик, -тель, -арь, -ун and many others.
They have some kind of rhyme and reason behind them, but it is so vague and there are so many ad-hoc exceptions, that you're better off just learning them by heart.
Even native Russian speakers get confused by distinctions like страховщик and страхователь, the legal names of the two parties in an insurance policy agreement. They are both formed from the Russian word страховать "to insure" by appending two different agent noun suffixes to them.
The suffix du jour for играть "to play" is -ок, hence the modern word игрок. As recently as 200 years ago it used to be -ец, which has fossilized in the set phrase и швец, и жнец, и на дуде игрец. And if you were into Church Slavonic, you would have used игратель (the suffix -тель is characteristic for Church Slavonic borrowings). Many did.
Видеоигра is quite a recent calque from English. Before that, Russian used the phrase компьютерная игра. "Video gamer" would hence be called компьютерный игрок.
There is the word видеоигрок, but it has not gained enough popularity yet, and I'm not sure it will.
but if "gamer" was translated to Russian, would it be видеоиграник or видеоигранец?
Since "game" is "игра", "gamer" should've been translated as "игрок". However, "игрок" is already kinda reserved for a "player" - a participant of a game - all kind of games: entertainment, gambling, sport etc, so that would be a bit confusing.
On the other hand, "gamer" majorly means a person that dedicates quite a significant time and effort to games, majorly computer and video games. So, that being a relatively recent phenomenon, the word "gamer" was borrowed verbatim from English to make a new Russian word "геймер".
As for "..играник" or "..игранец" - no, those two probably aren't the best way to form new words as those would sound rather foreign/incomprehensible and take some effort to decipher. But there are other rarely used variations like "игрун" and "игрец" that already been used and that you can put there with better success to indicate that you mean some new term, that you are joking, and/or that you aren't entirely approve of that activity.
As an example of how those words evolve: there is an ancient expression "и швец, и жнец, и на дуде игрец" meaning joe of all trades, a person that has to do a number of very different jobs that more recently appeared in a form like "Я тут работаю швецом, жнецом, и на дуде игрецом."