Is it possible transfer from a MS program into a PhD program within the same school?
This is usually kosher, but it can depend on the school and field. The department might want to see what you do over the first year or so, both academically and in terms of clicking with faculty and their research. The best thing for you to do now would be to find a faculty member whose research really interests you and try to find opportunities to work with them. If that doesn't work right off the bat, sign up for a class they teach and kick ass in it, then try again. Once you have this relationship, and assuming you're producing good work for them, it's rare that they wouldn't support you staying on, so long as they have the resources to fund you.

You, as a Masters student, are essentially a free trial for the school/department/professor. They get to see how you fit in the department and what you're capable of *plus* they don't have to pay for you. If you're a good fit, there shouldn't be a reason why they wouldn't want to keep you. However, they'll likely still ask you to apply (formalities).