Does an executor have authority to sell a family business if the will doesn't say so specifically?

Usually yes, but it depends on what the will actually says and what kind of business interest is involved. Most wills give the estate trustee broad general powers to sell, manage, or wind up estate property as needed, even without a clause naming the business specifically — an estate trustee is a fiduciary responsible for administering all the estate's assets in the beneficiaries' best interests, and selling an asset is a normal part of that job.

Where it gets more complicated is when the business interest is shares in a private corporation rather than a business run directly by the deceased. If there's a shareholders' agreement, it may include a right of first refusal, a required valuation process, or restrictions on who shares can be sold to, and the estate trustee has to work within those terms regardless of what the will says. The estate trustee also has to be able to show a sale was in the estate's best interest if a beneficiary later challenges it, which can come up when passing accounts. Reading the will's specific wording alongside any shareholders' agreement, before doing anything, is the right first step.