Can both terms be used to indicate a convenience store? I was having a discussion about convenience stores, and a chat friend sent 便利屋. I answered as convenience store. My answer was correct. The dictionary indicates a handyman. I wonder if the term could be a dialect or colloquial expression. Could the term be used as a literal translation? I am being downvoted so I need some clarification.
No, and I think you could have looked up 便利屋 in a dictionary before coining random words. 八百屋 doesn't deal with eight hundreds, nor department store is called 部門店．
便利屋 would roughly translates to odd-jobber; they deal with all sort of chores, including delivery of small amounts, fixing houses and so on. Originally they refer to the job as an occupation, but can also refer to any person who undertakes small jobs which others are reluctant to take.
For convenience stores, コンビニ or コンビニエンスストア are the only words that are understood.
and a chat friend sent 便利屋
as I mentioned above, 便利屋 doesn't mean convenience stores. However, it is a literal coincidence everyone can come up (when I learned about convenience as an English word, I thought 'So convenience stores are 便利-店; that's very suitable'.), so I think it's possible it was a kind of wordplay.
Could the term be used as a literal translation?
I don't think this question really holds --- I don't think I'd ever say "sure, it's usable as a literal translation but in reality it means another thing". However I do think many people who leaned the words convenience and store will be mildly interested in the coincidence.