Quantum Grassmannians?

apparently, quantum grassmannians come in many variations --- this may be what you are looking for:

Graded quantum cluster algebras and an application to quantum Grassmannians, Grabowksi & Launois, 2010.

Among those who study quantized coordinate rings, it is widely acknowledged that Grassmannians have a special place. The intricate geometric structures associated to Grassmannians, due in part to their Lie-theoretic origins, give a rich structure of their quantized coordinate rings, the quantum Grassmannians.

There is a deformation quantization approach to the quantization of the Grassmannians taking their Kähler symplectic form as the starting point. You can find this in the preprint by Schirmer arXiv:q-alg/9709021 from the nineties. He wrote a PhD on that. There are many earlier works on the quantization of $\mathbb{CP}^n$ in the same spirit, starting perhaps with works of Moreno and collaborators in the early eighties. You find many references in Schirmer's preprint.

However, this is not directly a $C^*$-algebraic approach, if you are interested in things like that. Nevertheless, the (a priori formal) star product is algebraic on many nice functions (the representative functions).