Do You Say Oftener?

"Often" sometimes has comparative and superlative "oftener" and "Oftenest", but forms with "more/most" are more common. So, both are correct, though "more often" is much more common than "oftener". I think you should use "oftener" if the text is very formal. Thanks a lot!

Is there such a word as oftener?

Oftener is not correct. The correct term would be more often. At least in the US.

What does the word oftener mean?

adverb. oftener; oftenest. Learner's definition of OFTEN. [or more often; most often] : many times : on many occasions : frequently.