Nope, they will not. You have to pay to a special account - MOSS account. Yes you have to add VAT - but not IVA. You have to add the local buyers VAT. Actually if your turnover is below 100.000€ - you can just base it on IP. Italian IP, Italian IVA foreign IP - foreign VAT. No need for address in that case. You could instead however also require address - and don’t look at the IP address (I guess most buyers want to have their address on their receipt/invoice - well and if they are Italian B2B I guess they want an e-invoice). I could not deduct an receipt over 100€ without my address on it. Below I could (not sure how it’s in Italy). Using either IP or address - you have to stick to a pattern and always use the same, plus record it for 10 years (same as invoices).
It’s two different accounts - one is your local IVA (monthly or quarterly depending on your turnover, maybe even yearly?), the other is the quarterly MOSS.
Also you have to issue invoices depending on the country - some countries require invoices, some only if the buyer asks for it. In Italy even e-invoices for B2B (but I guess you know about the dreaded e-invocing, which at least is not required AFAIK for B2C e-commerce transactions, Maybe also B2B e-commerce is excluded?). Additionally to the MOSS you have to state those sales on your esterometro - but that is Italian law, not EU law so a bit different. I’m moving to Italy next year so I’m just getting my head bummed by crazy Italian laws concering esterometro and e-invoicing (FatturaPA).
For normal invoices quaderno.io is a great solution, for FatturaPA / SdI I don’t know any good system working with s2member.
And no - just because your course is in Italian, you cannot assume all buyers are from Italy…
A good solution for VAT is to use Thrivecart for checkout instead of s2 pro-forms, that would check VAT number and automatically deduct IVA/VAT. Some user reported he got it working for s2member with a custom integration. Thrivecart will be good enough for you IF global turnover is below 100.000€
Not sure how good Thrivecart invoices are - they used to be shitty (meaning not legal).