Hello again Moreen,
If your dropshipper is located in the US while you're in the UK, then you are not liable to pay any sales taxes since you live outside the US. But if your dropshipper is based in the UK then you are likely to get additional VAT charges for your orders.
In turn when you sell the products in your store, unless you are VAT registered you cannot charge VAT for your products but you can transfer the VAT charges from your dropshipper to your customer. So in short you customer will pay the VAT charges of your dropshipper.
Considering if you should be VAT registered too? If your turnover is less then £60k a year you can be VAT registered if you like, however if your turnover is over £60k a year then you must be VAT registered.
I think you'll find this guide helpful - Link hidden: Login to view
Hope this helps, please let us know if we can be of any further help :)