The reason there's no cure for the common cold is that the common cold is caused by ~200 different viruses. You can vaccinate against the individual viruses, which prevents infection from that particular strain only.

There are additional complications in that many cold viruses generate only partial or temporary immunity, and the immunity wears off in a couple years. Also that coronaviruses mutate fairly quickly (though not as quickly as influenza, another illness that's caused by a family of viruses rather than a single one and generates only temporary immunity). I've heard that coronaviruses are also fairly hard to develop vaccines for because of technical reasons that I don't really understand.

But if your goal is partial, temporary immunity to protect against a circulating pandemic - similar to a flu shot - it's theoretically possible. That partial, temporary immunity is why adults get fewer colds than children do - they already have immunity to many of the cold viruses circulating within the local population.