Payments reach you through Stripe, and Stripe issues the salon's year-end payment form when the IRS threshold is met. It appears in your Stripe dashboard and is mailed to the address on the account.
Stylists you pay as contractors through LUX receive their own form for the amounts LUX paid out to them. Payroll settings in the app decide who is paid as a contractor and who as an employee; the salon payments policy sets out the rest.
LUX does not give tax advice. Keep the addresses and tax IDs in the app and in Stripe current before December, and ask your accountant what applies to you.