personally I'd go with 日本語研究会
grammatically both are fine. 部 is just the shortened version of the ブ in クラブ(when written in kanji 倶楽部).
For your situation, nihongo クラブ might sound better (maybe because most club in japanese have 3 syllables including the word bu, sound rolled off the tongue more. "nihongo" already have 3 syllables so it doesnt work as well. instead, If you read "Ni Ho N Go Ku Ra Bu" rhythmically correct, it should have 7 moras (rhythmic counter, different from syllables) exactly like in the haiku 5-7-5 rhythm.
Other choice, 日本語研究会 like @Fredora said is good too, sound official and dignified. Although in Japan, 会 has a slightly different meaning: club (and being able to call your club "club") you need to register with the school and have a supervisor teacher, while "会" or "society" dont have that requirement, meaning you can create as many societies as you want and just meet anywhere you want, but you dont get to have an assigned personal club room and a club allowances to buy equiments for club activities.
部 versus クラブ (for saying "Japanese Language Club")
I'm starting a Japanese language club at my school. Do I use 部 (for nihongo bu) or クラブ. I think the second is best so my students can recognize the characters and feel proud, but I also want to be accurate.
Thanks in advance !