I am married under the regime of community of property reduced to acquisitions. My husband has inherited an undivided share in a real estate property. Today, we wish to purchase the share of his co-owner. Will I become the owner of the property?

No. Under the community of acquisitions regime, assets received by way of succession are considered separate property of the spouse who inherits them (Article 1405, paragraph 1 of the French Civil Code).
Consequently, the undivided share inherited by your spouse is his separate property. If you purchase the share of his co-owner, the law provides that it will also remain the separate property of your spouse, subject to his obligation, upon dissolution of the community (death or divorce), to compensate the community (Article 1408 of the French Civil Code).