The Periapt of Wound Closure's description says

While you wear this pendant, you stabilize whenever you are dying at the start of your turn. In addition, whenever you roll a Hit Die to regain hit points, double the number of hit points it restores.

Emphasis mine.

What happens to a PC wearing Periapt of Wound Closure while suffocating? I'd like to transport myself using a Homunculus and a Bag of Holding, but I worry about suffocating during the journey.

If I put on a Periapt of Wound Closure and enter a Bag of Holding, will I suffocate to death while inside of the bag or will the Periapt keep me alive?

You would suffocate to death unless it is a short trip.

Bag of Holding:

Breathing creatures inside the bag can survive up to a number of minutes equal to 10 divided by the number of creatures (minimum 1 minute), after which time they begin to suffocate. roll20

Suffocating (emphasis mine):

When a creature runs out of breath or is choking, it can survive for a number of rounds equal to its Constitution modifier (minimum of 1 round). At the start of its next turn, it drops to 0 Hit Points and is dying, and it can’t regain Hit Points or be stabilized until it can breathe again. roll20

Thus, the Periapt of Wound Closure can not stabilize you. You'd have to start making Death Saves after you drop to 0 HP. Unfortunately, three successes can't stabilize you either, so you'd be making Death Saves until you died or could breath.

Note: The PHB errata updated the wording of Suffocating to include that you can not be stabilized.