Currently there is no dual degree program for mechanical engineering with finance The only dual degree program is of BS-MS for software engineering in vit Vellore.SE / IT / CSE are the strong streams of VIT if you are interested in these courses you can apply directly to college from website or they do offer dual courses of BBA and mba for finance you can think of that too.

If you are interested in both fields you could first complete a B.Tech in Mechanical Engineering and then pursue an M.B.A with a finance specialization.