Would it be wise to ask my current undergraduate research advisor to put me in contact with the people he knows within the math department at his former Ivy league school. I really need help to best navigate the academy.
What you suggest seems fine to me, but you could, and probably should, take it a step farther. Try to arrange a series of conversations with your advisor about how to approach graduate school in the long term and the admissions process in the short.
In particular, you will learn a bit about what life is like for a student at the institution you want to go to. It won't all be rosy and it will be hard.
He probably also knows a bunch of people other than at his alma mater and could make recommendations about them as well.
The personal touch never hurts, but with some admissions systems it may not help a great deal. The admissions process is often handled by a committee or even by administration rather than by individual professors. You will still need to make a good overall impression on the admissions people in any case.
On the other hand, don't necessarily close down your options to a single person. If that doesn't work you need the flexibility to go elsewhere.