After a purchase has been made, you can either send the other member a conversation on Etsy or email them directly.
Go to You > Purchases and reviews and after locating the order, click Contact The Shop on the right.
If you use guest checkout, reply to the receipt email that was sent to you after purchase.
Go to Shop Manager > Orders & Shipping. Then find the transaction you're looking for and click the member's name at the top.
Click Contact to open up a new conversation or click their email address to open a new email.
You can also email the member, which is in the transaction notification email Etsy sent you at the time of purchase. Look for an email from firstname.lastname@example.org.