Virginia does not have an inheritance tax (i.e. a tax paid by the beneficiary on amounts received due to the death of the previous owner). Rather, Virgina does impose an estate tax.
The Virginia estate tax is a transfer tax on the estates of resident decedents and the estates of nonresident decedents having real estate or tangible personal property located in Virginia. Estates of alien decedents having real estate or tangible personal property located in Virginia, as well as intangible personal property attributable to Virginia are also subject to the tax. Only those estates that are required to file a federal estate tax return will be subject to the Virginia estate tax requirements (i.e. estates of decedents whose fair market value exceeds $2,000,000). Virginia will only issue a "certification of payment" after a VA estate tax return has been filed. Accordingly, if no estate tax return is required to be filed, the VA Dept. of Revenue will not certify that no estate (or inheritance) tax is due as this would likley preclude their right to assess such tax in the future.
The personal representative/executor of an estate must file a Virginia Estate Tax return (Form EST-80) whenever required by federal law to file the federal estate tax return.
What form are you reviewing where this is required?
Because it is impossible for me to identify and consider ALL the relevant facts, this advice is not intended or written to be used for the purpose of avoiding penalties, and cannot be used for that purpose.
I would call the transfer agent or custodian of the shares and ask them what the procedure is when there is no inheritance tax to be waived. The verbage you quoted even allows for the possibility that there may be no inheritance tax (i.e. "the financial institution may advise you that an Inheritance Tax waiver Form is required"). The number should be in the transfer paperwork. The answer may be that you can ignore the requirement.