Can an object have momentum without speed?

No, it cannot. In the case of an object moving in the free space (no forces acting on the object) the energy consists of only the kinetic energy which is proportional to squared momentum. Thus, if the object has a momentum it has an energy to. Basically an object possesses some energy in any kind of time, and it might happen that the energy is zero. It doesn't mean that it has no energy. It means that the object has energy equals zero (which is not the same).