WASHINGTON—Family-owned businesses including grocery stores, craft shops, small manufacturers and others are worried tax legislation in Congress could leave them at a disadvantage to big corporations and other competitors.

At issue for these businesses is their structure as trusts, established to preserve an enterprise for succeeding generations, protect against estate taxes or a divorcing spouse or other claimants who might try to seize a stake. Some family-owned firms say the Senate version of the tax bill holds risks for...