For US Sales Tax and compliance we can provide guidance only, but we cannot file or advice in an official capacity as it not a UK or EU tax. We speak about this topic regularly, and can provide some helpful details on considerations and the most common technical solution used by our clients, the most popular solution being TaxJar*. This is fast-moving area of tax law, and tax legislation has not yet caught up with how ecommerce has changed what selling looks like in practice.
As a quick primer on the topic: the rules are complex, more so than with VAT, as Sales Tax varies by product, State (and sometimes city within the State), where your customer is, and where your product is shipped from. The test for whether you have to file in a state is typically based on whether you hold inventory in that state (for example if you are Amazon FBA in the US) or if you have a certain level of economic activity in a state, with most thresholds starting at $100,000 revenue in a year or more than 200 orders (these thresholds vary and one state has zero threshold).
As a result the cost of compliance in the US can be in the region of $25,000-$40,000 if you are active in each state. Lower if you use a software like TaxJar, but still far higher than handling VAT. For this reason, the reality is that many businesses with lower revenue will opt to ignore their Sales Tax obligations on the basis that the risk they face of non-compliance is lower than the cost of compliance. Larger businesses, with greater revenue (i.e. greater tax liabilities and potential fines), and more to lose, who can afford this compliance cost simply view this as a cost of doing business.
*This is an affiliate link which earns you a discount and us a referral commission. Here is the the unaffiliated TaxJar link.