Learn French: L'actionnaire rejette formellement ce rachat. - The shareholder formally rejects this takeover.

You would remove the adverb formellement and use ne... aucunement or ne... nullement around the verb: L'actionnaire ne rejette aucunement ce rachat. Remember that these expressions are absolute negations, so you never use pas alongside them.

What exactly does rachat mean, and how does it relate to acheter?

Acheter means to buy, and adding the prefix r- generally means doing it again. While racheter can simply mean to buy back or buy again, in the corporate world un rachat specifically translates to a corporate buyout or a takeover.

Is the word actionnaire related to the English word action?

It is, but with a very specific business meaning. In French finance, une action is a share of stock in a company. Therefore, un actionnaire (or une actionnaire for a woman) is a shareholder.

Why is formellement placed right after rejette?

In French, an adverb that modifies a conjugated verb is almost always placed immediately after that verb. While English often places adverbs before the verb (formally rejects), French keeps the adverb anchored directly after the action it modifies.