No. They are two different registrations issued for different purposes. A VAT number identifies you for value added tax. An EORI number — Economic Operators Registration and Identification — identifies you to customs authorities when goods move across a border. Having one does not give you the other.
They can look related, because a GB EORI number for a VAT-registered business is normally built from the VAT registration number with a country prefix and a suffix. That is a formatting convention, not the same identifier, and you still have to apply for the EORI separately before you can use it.
You need an EORI to import or export goods. Businesses that are not VAT registered can still get one. Depending on where you move goods, you may need more than one — for example a separate EU EORI if you are the declaring party there, and separate arrangements for movements involving Northern Ireland.
This matters for asset finance more often than people expect. If you are importing machinery or a vehicle, customs will not clear it without the right EORI, and import VAT and duty have to be dealt with before the asset is released — which affects both your delivery date and when funds are needed. Your freight agent or customs broker should confirm what is required for the specific import, and your accountant should confirm the VAT treatment.
This is general information, not tax advice. Confirm your own position with your accountant or HMRC.