Not many really and they're more lexical than grammatical...
a) My tutoring company has decided to start a marketing team ...
Better : to set up a marketing team
an action and also some organisations like companies, but set up
a group of people.
b) Part of our plan on the Facebook page is to introduce our tutors...
The "plan" isn't on the FB page, so I'd change this to Part of our plan for this Facebook page is to use it introduce our tutors ...
or even better : One of the things we intend to do on the Facebook page is to intriroduce our tutors...
c) brief tutor introductions can enhance the center's personality
A centre can't have a personality. Better : ... can enhance the center's image...
d) cartoon images
: This makes it sound as if they'll be caricatures or funny drawings? Is this what you mean? It has a very negative connotation. I suspect you just mean drawings. Compare this caricature
of Obama with this drawing
e) to cater to those
: to cater for those...
can have the connotation of reluctance, of not really believing it should be done : Their father is a tyrant. They have to cater to his every whim.
f) uncomfortable with their pictures being on the Internet.
But if you put drawings up, then these are also "pictures". Here, I think you mean "photos".
g) In the next week or two, the marketing team will be taking pictures
- again "photos"
h) of everyone to transform into cartoons. Transform
is a transitive verb so ...and will transform them into drawings.
or ...into pictures
Hope that helps.