If you want a raise with your next contract, you need to talk to your employer now. In this case, your employer is the "vendor". You just speak with your boss at the vendor and say "This contract is coming up for renewal in a few months. I really like working there, but for my next contract, I would like $X."
Usually, a company has a budget in mind when it seeks out a contractor through a vendor. That rate isn't always firm, so it might be possible that they would be willing to pay more. At the same time, there are many contractors out there - probably available through the same vendor.
You are in a position where the company knows your work. If they want to extend your contract, that indicates that they like your work, and probably don't want to bring in a replacement contractor. That works in your favor.
When I have hired contractors in the past, I had to get the contract approved by folks higher up in the corporate ladder. It was fairly simple to get the contract extended - as long as the rate didn't change. But if I wanted to increase the rate, I would have to go through another requisition process, and approval process, which was the equivalent of starting over. That meant I always tried to renew at the same rate. That works against you.
Asking your vendor for a higher rate could work. Either the company could pay a higher rate, or the vendor could accept a smaller percent of your rate and thus pay you more. Or, your need for a higher rate could mean that this particular position is no longer viable for you, and the vendor must find you a different position. You might find yourself on the bench for a while, waiting for the higher rate position to come along.
To sum up, if you want a higher contract rate - start with the vendor you work through.