You can avail loan for any university and for any amount less than Rs. 10 Lacs from any bank given the bank is ready to offer the loan for the said university.
For VIT almost all banks will be ready to offer you loan. But as far as management quota is concerned, if in the admission letter the full amount is mentioned i.e whatever is the fees for management quota should be mentioned in admission letter, then the bank will consider that amount and will grant you the loan below 10 lacs.
Thanks.