Talk to a lawyer. I heard Ramesh Shrestha is good in business related cases.

First of all, share which state is it?
And check for its non-competition and non-solicitation laws?
I had a similar case in California where I was threatened but when I checked the law, I found out such terms are null and void unless it has to do with trade secrets.

The consultancy is in NJ and the vendor and client are in Ohio. What I'm asking is if consultancy agreed with vendor on 3 months contract to hire, they agreed that I could be employed by client after 3 months. So can they still ask for damages from me based on having to work with them for 12 months contract I signed with them?

I don't think so. If you also have a copy to prove that they agreed with vendor then you will win it.
Tell them to bring it on.

Just saw agreement between consultancy and vendor, it seems they had agreed to 6 months contract. The job description and vendor told me its 3 months to hire and I was hired after 3 months. So now does the vendor have to pay the damages too?

Since the vendor and client are in Ohio, are the agreements you signed based on Ohio or NJ's laws? NJ can't enforce laws on clients who are not in state.

Reviewing Ohio's non competition laws, it doesn't seem enforceable in your case.
https://www.ohiobar.org/forpublic/resources/lawyoucanuse/pages/lawyoucanuse-357.aspx

T-mobile it does seem enforceable reading your link to me

As "T-Mobile" already suggested asked them to "bring it on". They cannot do shit about it. They are just trying to scare you.. and here's why they cannot do shit about it..you can counter sue them as well.