No. There are a number of abilities that come close, but there is currently no official method to add your proficiency bonus to the Dexterity check you make for initiative.

• The Barbarian gains advantage on initiative checks with Feral Instinct
• The Bard adds half their proficiency bonus to initiative with Jack of All Trades
• The Fighter (Champion) adds half their proficiency bonus to initiative with Remarkable Athlete
• The Rogue (Swashbuckler) adds their Charisma modifier to initiative
• The Ranger (Gloom Stalker) adds the Wisdom modifier to initiative
• The Wizard (War magic or Chronurgy magic) adds the Intelligence modifier to initiative
• The Enhance Ability spell can give advantage on initiative with the Cat's Grace option
• The Alert feat gives a +5 bonus to initiative (note that this is as good as or better than proficiency at all levels before 17)
• The Rod of Alertness gives advantage on initiative checks
• The Sentinel Shield gives advantage on initiative checks
• The Weapon of Warning gives advantage on initiative checks

Different AC calculations don't stack; you pick which one to use

As Slagmoth pointed out, this general topic has been asked and answered many times by the designers, and an official answer appears in the Sage Advice Compendium:

These [AC calculation] methods—along with any others that give you a formula for calculating your AC—are mutually exclusive; you can benefit from only one at a time. If you have access to more than one, you pick which one to use. For example, if you’re a sorcerer/monk, you can use either Unarmored Defense or Draconic Resilience, not both. Similarly, a druid/barbarian who transforms into a beast form that has natural armor can use either the beast’s natural armor or Unarmored Defense (you aren’t considered to be wearing armor with natural armor).

(The entry goes on to clarify that AC bonuses, like those provided by shields, do stack with AC calculations; it's simply that if your features give you different ways to calculate your AC, you pick one.)

In addition, Keith Baker - original Eberron designer, and primary author of the Wayfinder's Guide to Eberron - has posted an FAQ on his website about the Wayfinder's Guide (which he has continued to update to answer questions). It contains a section about the Integrated Protection feature, which specifically answers this question and references the Sage Advice ruling:

If I’m playing a warforged barbarian, do I get to add my Constitution to my Integrated Protection? What about a warforged monk?

No and no. Integrated Protection and Unarmored Defense are two separate features that set your AC, and the Sage Advice ruling on this is that you pick one—you don’t combine them. If you have your Integrated Protection mode set to darkwood core, you are considered to be unarmored and can thus use Unarmored Defense—but you’re always either using one or the other to determine your AC.

The section also clarifies whether the Integrated Protection feature counts as "armor"/"being armored" for other features. It's a worthwhile read.

In short: you have the choice between two different AC calculations:

• The warforged's Integrated Protection: 11 + your Dexterity modifier (+ your proficiency bonus, if proficient with light armor)

or

• The barbarian's Unarmored Defense: 10 + your Dexterity modifier + your Constitution modifier

They don't stack. You simply have the choice between one and the other (presumably you'd choose the one that gives you a higher overall AC).