No. Dependent status in the L category is available to a legally recognised spouse and to unmarried children under 21, and a common-law or de facto partner does not qualify as a spouse. A partner in that position needs an independent status of their own, which will not be granted because of the relationship.
The relationship has to be one American law recognises as marriage
Couples who have lived together for years, hold property jointly and are treated as partners for provincial purposes are often surprised by this, and it is better discovered now than at a consular window. The dependent category follows marriage, and provincial recognition of a common-law relationship does not create one for these purposes. Where a couple intends to relocate together and both need to work, the realistic options are to examine the partner's own eligibility on its own facts, or to consider marrying before the filing, which is a personal decision with legal consequences and is not something a site can advise on.
If the couple does marry, the spouse admitted in L-2 is employment-authorized incident to that status and may work for any employer. Children under 21 who are unmarried may accompany them and may attend school, but may not work, and each child's twenty-first birthday ends that eligibility.