Yes. Where you find them depends on your role.
Beneficiary
Sign in and go to supportnow.org/gifts (Your gifts). Every gift sent to your Support Page is listed with who sent it, the amount, and their message. Unopened gifts show as Sealed. Use Open on one gift, or Open all.
Your dashboard also shows a "gifts waiting for you" notice when any are unopened, and a Gifts section listing each one.
Every gift is already part of your Support Page balance and pays out with your other donations. There's nothing to claim.
Organizer
Open your Organizer Dashboard and go to Donations. Each gift has a Gift badge, the Gifts filter shows only gifts, and the totals at the top show how much was Sent as Gifts, how much has been Opened, and how much is Not Yet Opened. The gift email goes to the page's beneficiary, not to organizers.
Supporter who sent a gift
It's listed with your donations on the Supporter tab of your dashboard, marked Gift. If the family sends a thank-you, it reaches you as a message from them.