If temperature makes particles vibrate faster, and movement is limited by the speed of light, then I would assume that temperature must be limited as well.
Why is there no limit?

I think the problem here is that you're being vague about the limits Special Relativity impose. Let's get this clarified by being a bit more precise.

The velocity of any particle is of course limited by the speed of light c. However, the theory of Special Relativity does not imply any limit on energy. In fact, as energy of a massive particle tends towards infinity, its velocity tends toward the speed of light. Specifically,

$$E = \text{rest mass energy} + \text{kinetic energy} = \gamma mc^2$$

where $\gamma = 1/\sqrt{1-(u/c)^2}$. Clearly, for any energy and thus any gamma, $u$ is still bounded from above by $c$.

We know that microscopic (internal) energy relates to macroscopic temperature by a constant factor (on the order of the Boltzmann constant), hence temperature of particles, like energy, has no real limit.
