We do sales tax, VAT, and other indirect taxes in-house. At our scale, external providers are too expensive.
At my last job, we handled worldwide VAT and GST compliance in-house, and left sales compliance tax to Avalara. Most countries insist that you begin to file as soon as you have any sales to their country. As a practical matter, you don't need to register unless (a) you're doing at least $1000 in taxable sales quarterly or (b) the tax authority reaches out to you and tells you that you need to register.
VAT and GST compliance is very straightforward in most countries: it's a 1 screen form. Payment is also easy, but can be pricey: wire transfers. Best to have your accountant do them, but if he's terrified of the VATMOSS, you need a new accountant. The VATMOSS is an extremely simple form (and the VATOSS, for larger companies, isn't much more complex).
Sales tax compliance is hard, because the U.S. has over 10000 different sales tax districts and more than a hundred sales tax jurisdictions (but you probably won't have to file in most of them). Avalara is a bit pricey for the returns ($50/each) so you may want to do those in-house or have your accountant do them.
NOTE: I DO NOT RECOMMEND AVALARA. They were purchased by a PE company and the quality of service plummeted off a cliff even as prices tripled. Use literally any of their competitors instead.