Do I require an IRS Number or SSN when shipping to the USA?
When importing into the USA, you may be required to provide an IRS Number for the ultimate consignee.
This is outlined under CBP 3550-079A "Ultimate Consignee at Time of Entry Release"
https://www.cbp.gov/document/directives/3550-079a-ultimate-consignee-time-entry-release
"For formal entries, the appropriate identification number for the Ultimate Consignee is defined as an Internal Revenue Service employer identification number, or a Social Security number"
Do I really need to provide the employer identification number or Social Security number?
The short answer, is, yes. If you are selling directly to the USA and to an end buyer, these numbers are required.
How can I avoid the hassle of trying to provide the EIN or SSN?
One way to handle this is to ship the goods, unsold, to a USA warehouse or pick & pack operation.
The USA warehouse can provide their EIN, and then when you ship your items from the USA warehouse, there is no EIN or SSN requirement.
A & A can help you find a warehouse / distribution center to best handle your pick & pack.