One great idea for storage is underground basalt formations. Inject CO2, and it'll turn into limestone in under two years. It can't solve the whole problem by itself, but there's enough capacity for gigatons of CO2.
There are several entities working on this. Here's a company with a small pilot project, and links to scientific papers: https://www.carbfix.com/
Artificial production alone doesn't do the trick. It would have to be low in energy consumption, preferably exothermic. I don't think there is such a reaction with plentifully available material. Otherwise you spend more energy on the basalt than the CO_2 production gave you.
Also, large parts of this planet consist of natural basalt deposits, every area with some current or historic vulcanism has them. E.g. half of Siberia is a huge basalt deposit. Google Siberian Trap. So I do not think artificial production would be necessary.
There are several entities working on this. Here's a company with a small pilot project, and links to scientific papers: https://www.carbfix.com/