Since the first mission is likely to be a simply Apollo 8 style Lunar fly-around/free return; the answer is almost certainly yes. You can do it with a reinforced & slightly reconfigured Soyuz/Shenzhou with a Progress-derived habitation module linked; & an off the shelf kick stage with strong Proton or Long March heritage.

What will happen next is the interesting bit: the PRChinese are looking to the Moon as their first major goal; but they are also investigating seriously orbital dynamics around small bodies, which suggests they may consider a nearEarth asteroid mission to further stretch their technological envelope while developing the essential lunar lander technology. The PRC will also exploit existing ties with Japan, Europe, India & Russia to help defray the cost of this program - all of whom would more happily buy in on a cheap lunar/asteroid program than an expensive NASA Mars concept (especially given NASA's increasing financial unrelaibility as a a big project leader).

One of the more entertaining futurist possibilities is that the next manned mission beyond Earth orbit will be an international venture with little or no USAmerican involvement - partially because of the usual Washington paranoias about the PRC; but mostly because the residual Mars clacque will prevent the US from buying in on a lunar/asteroid mission without a preposterous investment in the Red Planet as well. No Mars/no Moon won't end with Mars plus Moon in this case; but no Mars or Moon for the children of Apollo....