Vellore

Yes, it is available in VIT. Even my friend has taken that course.

The eligibility criteria for VIT-Vellore is a minimum of 60% marks in an undergraduate degree (minimum 3 years duration after 10+2) from a recognised university/institute.
Only the candidates who have studied in regular full-time programmes are eligible to apply.
After applying for the college, you will be called for a written assessment test (WAT) and a personal interview (PI).
The final selection will be made based on the merit list using a composite score with the following components:
Academic Performance
Profile-Work Experience, Awards and Recognition, etc.
WAT & PI score
Management Aptitude Test Score.

Mechanical branch in VIT, Vellore gets filled by a rank of around 10k. In VIT, Chennai it's available till a rank of around 15k. Marks around 90 for Vellore and 75 for Chennai.
P.S- It is based on my experience during counselling.

VIT MBA Admission offers two-year MBA and MBA (IB) programmes which consists of six trimesters. Students can choose a range of elective subjects from areas such as finance, marketing, human resources, information technology and systems, international business, operations and business analytics. The MBA program is offered out of the Chennai as well as Vellore campuses. The VIT MBA admission 2018 in International business is available only at the Vellore campus.
Refer this link: https://www.shiksha.com/mba/course/m-b-a-in-international-business-vit-business-school-vellore-242303

Alright, where do I start? Well I have been part of VIT for long now. VIT is really doing good in B.Des/B.Tech here and shapes your career in right direction. VIT gives you chance to grow as potential students and lateral as an entrepreneur or employee. Although VIT cannot match the salaries and packages of the IIT's or even some of the top tier NITs but can assure you that the placement cell here gets all the "high flying" companies to campus and you're assured a good job. Placements happen throughout the year and people get up to 3 high paying offers. This is one of the selling point for VIT, guaranteed job after four years.

You have to pay 1,76,000 INR. (1,73,000 tuition fees+3000 caution money) for the first year then 1,73,000 for all other years if you are from the first category, otherwise, you have to pay more if you are from the higher category.

VITEEE is an engineering entrance exam which is for only engineering courses. For integrated M.Sc in Biotechnology course, there is no entrance exam and the admission is based on board exam marks.

First, congratulations to you. You should differently get VIT -Vellore campus, if you choose any branch.

For VIT Vellore the VITEEE entrance test is held in the month of April but sorry to say the Exam is over.

There are plenty of colleges which offer lateral entry in India but not possible in VIT. I don't remember when there was lateral entry last time in VIT. Very similar to VIT one college SRM do offer lateral entry so you can try your luck there. There are less number of seats in SRM so you need to apply at the earliest.
Hope this helps.