I don't think this can be answered unless we know what country you operate in.
Even then, you should primarily consult local accountants and tax advisors.
Anyhow, I think many companies either operate in countries which have lax laws, or they don't care too much.
If you need invoices for the payments you send, it will be quite hard to get invoices from affiliates... a proper solution could be self-billing invoices I believe, which is used by Hilton for affiliate commissions, and LiveJasmin for performer commisions apparently.
Some links, all EU specific:
VAT: self-billing arrangements - Detailed guidance - GOV.UK
VAT invoicing rules - European Commission (self billing is mentioned)
Self-Billing Agreement for UK Travel Agencies - Hilton
Periodical Self-Invoices - Easy Wiki - Jasmin Wiki
I'm not too knowledgeable about US stuff, but some sponsors make affiliates fill out W-9 and W-8 forms. More common in mainstream than adult though.