I (living in Germany) bought a laptop for my cousin (living in Bosnia). - Can the tax be reclaimed from export to Bosnia?

If yes how? Have never done something like this.

Products were purchased through my Amazon account and addressed to me. Now this is picked up personally by my cousin, since she is just nearby.

The laptop was bought by you on your behalf and delivered to your domestic address. This is a VAT-liable delivery. If you then pass on the laptop to a person living abroad, this is your private income and can't lead to the VAT!

It would be different if you yourself are an entrepreneur and sell the laptop to a person abroad!

Yes - but you have to confirm the export at the border.

For a value of more than 1000 EUR you have to create an export accompanying document.

The best you call times at the Amazon support and ask what they need exactly.

If the VAT on the bill your cousin expelled, you can get them back with the stamp of the Bosnian customs