Are online gifts receipted automatically?

They can be. Set an Auto-receipt value in Settings → Receipts, and online gifts at or under it from per-gift donors are receipted and emailed on arrival.

They can be. An Org admin sets the Auto-receipt value (CAD) in Organization Settings → Receipts. When an online donation comes in at or under that amount from a donor whose Tax receipt preference is Per gift, Donors issues the receipt and emails it with the PDF attached. A note is added to the gift saying the receipt was auto-issued.

Details

Auto-receipting is skipped, and the gift waits as Needs receipt, when:

The gift is over the auto-receipt value, or the value is $0.
No authorized signature is on file.
The donor's name, street, city, province or postal code is missing, or they have no email.

In those cases, and for donors set to Consolidated annually or Do not receipt, the donor gets a thank-you email without a receipt. Donors set to Consolidated annually are receipted in the year-end run. Gifts you record by hand are never auto-receipted; you generate those from the gift or the Receipt center.