Vat Recording & Reports for Multiple Vat countries

Hi Jessica S1 Thanks for reaching out to us here on the Community. It is possible to track separate VAT/tax returns by creating different agencies as you've alluded to in your post: 
Go to Taxes from the left menu bar and select Add tax > Custom tax > enter the details > Select Save. 

If you would like to create tax codes specific to the different agencies then you would select the relevant agency within the tax agency field when creating a new tax rate: Taxes > Add tax > tax rate > enter the details > Save.

When you then create transactions, and select either exclusive or inclusive in the amounts are field, you'll be able to select the Tax rate that you've created in the VAT column.

I have also included an article that references the above: https://intuit.me/3WBMUww

I'm unable to provide information regarding if Shopify would facilitate the above however there maybe a Community member that may view this post and offer an insight, you could also reach out to the Shopify support team.

Yes, all this is possible by adding separate agencies and rates.

In terms of integration, you need to use a third-party integration tool such as A2X accounting to ensure everything is mapped correctly and accounted for. I am currently using it for a client with very similar sales tax complexity and it's working smoothly.