Is there anyway I can get my mom a Coronavirus vaccine while she's in the hospital?

That would be a question for her doctor, the hospitalist, or the hospital itself. Ask for a patient care manager or social worker and speak with them. Good luck. Do know that they now say that you can be several weeks late with the second shot and it is just fine. I wasn't aware that it was as long as cwillie says below, but it is good to hear that is the case. Truth is that they are still researching as we go and we may have 85% protection from just one shot. But so far the two is the recommend.

No need to panic, in Canada the science nerds have recommended to extend the interval between the first and second dose of Moderna to 16 weeks.

Good question. Can you not ask her Dr if perhaps she can get the second one while she's there? I'm sure Dr's have access to the vaccines, or at least I would hope they do. Good luck with that.