it seems that polymetallic mining from the seafloor is inevitable
This is incorrect.
Are there other sources of these elements that can be used other than the seafloor?
You're saying this as if we are currently mining seafloor deposits and nothing else.
The other answer provides a comprehensive list of the current production of Ni, Co, and Mn. In short: There's plenty more Ni, Co, and Mn around and there is absolutely no shortage of us running out of that in the coming hundreds of years. On land. Not in the ocean. http://elementsmagazine.org/past-issues/deep-ocean-mineral-deposits/ Most of the stuff is paywalled, but finding the texts online isn't too hard.
Specifically regarding seafloor mineral deposits, Elements Magazine published an issue two years ago outlining some of the prospects, issues, challenges, and promises in the field of seafloor mining.
One of the greatest problems is the enormous environmental impact. If you're mining on land, the environmental impact is pretty much localised to where you are mining. Once you're in the oceans, you're disturbing many kilometre-squares of area. With environmentally conscious consumers and companies, there will be no market. Volvo, for example, are pledging for ethical sourcing of their metals. They will not buy anything produced from the seafloor. Others have followed, or will follow soon. In that case, if there's no market, there's no need for production.