I got an 88 in a class this semester, but my professor gave me a B instead of a B +. It is my school's policy to to award + - grades, so my question is supposing my professor keeps my grade at a B for some reason, would LSAC change it to a B+ come time to convert the grades?

Lsac will calculate what it says on your official transcript.
If it says B it will be calculated as so.
Most schools leave it up to the discretion of individual professors to give +/-.
Check the syllabus, if it says 88 is B+ talk to the professor or take it to the dean of your college if necessary. However I would bet your professor stated that he doesn't award +/- which means your professor doesn't have to give you one.

Best thing to do in either situation is to politely talk to your professor. Explain how it makes a difference to lsac, if he thinks you deserve the + then maybe you'll get it

the syllabus says he gives + - for 8-9 and 1-2. So I'm good right?

Contact the prof telling him/her that your grade is supposed to be B+ based on the syllabus

TITCR

Be polite