Thanks for the reply and the links! Everything you said and assumed is correct, and I was afraid that mbstring is necessary to use Mibew. When I do phpinfo on my system, I have three lines that mention Multibyte, and all are similar to this one:
Zend Multibyte Support disabled
I have tried to follow the info in the second link that you posted. I was able to install php70u along with php70u-mbstring, but I could not get it to work with Mibew. I'm not sure where the incompatibly lies (with Mibew, Apache, PHP, etc.) but something wasn't working correctly. I would also like to keep with the current PHP version supplied by yum, it makes updates way easier.
I'll keep searching for a solution, I'm just not sure that there's one out there that will work for our situation.