Customs oldtimer

Firstly an EORI is an importer/ exporter registration number to allow an entity to be recognised when making out an import or export declaration.
There is no requirement to have a VAT number to obtain an EORI but if the entity is VAT registered the EORI and VAT numbers are linked.
The EORI itself places no direct financial or reporting obligation on the entity.
There are however responsibilities for ensuring the accuracy of customs declarations even if these are made by someone else on behalf of the entity. https://www.gov.uk/guidance/due-diligence-when-making-customs-declarations I would expect as a charity you will have officially appointed trustees and usually someone specifically nominated for the finances who can apply on behalf of the charity.
Those authorised persons can also appoint someone within the organisation (in writing) to make the application on behalf of the organisation.

The £135 threshold is for the goods themselves. So without the VAT and shipping charges.

As the UK is no longer in the EU the intra EU community distance sales rules do not apply to UK businesses. Instead Import VAT applies at the local rate of the country the goods are sold to. You zero rate UK VAT as the goods are not sold to UK customers.

Hi SJM The value of the goods is £ 68 The retailer is adding the UK VAT onto their sale price as the value of the goods is below £135. Provided the seller quotes the correct details, cost of goods and the shipping and VAT separately plus their UK vat number there should be not be any additional charges.

Please check if your supplier is using DDP Incoterms or a carriers DDP service as this can make a difference who is responsible for what charges and responsibilities apply. DDP may not be the best choice if you are the named importer. Only the importer and owner of the goods can reclaim UK import VAT paid. Most UK VAT registered companies however choose to use postponed VAT accounting meaning that VAT is accounted for via your VAT return instead of paying at import. The same rules about reclaiming applies. https://www.gov.uk/guidance/vat-imports-acquisitions-and-purchases-from-abroad You will need to download either a C79 if VAT is paid or your PVA monthly statement via your government gateway account as proof of import VAT paid/ due. https://www.gov.uk/guidance/manage-your-import-duties-and-vat-accounts

To add there is further legislation here: www.legislation.gov.uk/ukpga/2018/22/section/21